Low density parity check code for terrestrial cloud broadcast

ABSTRACT

Provided is an LDPC (Low Density Parity Check) code for terrestrial cloud broadcast. A method of encoding input information based on an LDPC (Low Density Parity Check) includes receiving information and encoding the input information with an LDPC codeword using a parity check matrix, wherein the parity check matrix may have a structure obtained by combining a first parity check matrix for an LDPC code having a higher code rate than a reference value with a second parity check matrix for an LDPC code having a lower code rate than the reference value.

CROSS-REFERENCE TO RELATED APPLICATION

This application is a Continuation of U.S. application Ser. No. 13/906,843, filed on May 31, 2013, and this application claims priority to Korean Patent Applications No. 10-2012-0059089 filed on Jun. 1, 2012 and No. 10-2013-0055762 filed on May 16, 2013, the contents of which are herein incorporated by reference in its entirety.

TECHNICAL FIELD

Embodiments of the present invention are directed to LDPC (Low Density Parity Check) codes for terrestrial cloud broadcast signals to correct errors that occur on a radio channel in a terrestrial cloud broadcast system that operates in a single frequency network.

DISCUSSION OF THE RELATED ART

The current terrestrial TV broadcasting causes co-channel interference that amounts to three times the service coverage and thus cannot reuse the same frequency in an area that is within three times the service coverage. As such, the area where frequency cannot be reused is referred to as white space. Occurrence of a white space sharply deteriorates spectrum efficiency. This situation led to the need of a transmission technology that facilitates reuse of frequency and removal of white space, which focus upon robustness in reception, as well as an increase in transmission capacity, so as to enhance spectrum efficiency.

A terrestrial cloud broadcast technology that provides for easy reuse, prevents white space from occurring, and allows a single frequency network to be readily set up and operated has been recently suggested in the document entitled “Cloud Transmission: A New Spectrum-Reuse Friendly Digital Terrestrial Broadcasting Transmission System”, published IEEE Transactions on Broadcasting, vol. 58, no. 3 on IEEE Transactions on Broadcasting, vol. 58, no. 3.

A use of such terrestrial cloud broadcast technology enables a broadcaster to transmit broadcast content that is the same nationwide or different per local area through a single broadcast channel. To achieve such goal, however, the receiver should be able to receive one or more terrestrial cloud broadcast signals in an area where signals transmitted from different transmitters overlap, i.e., overlapping area, and should be able to distinguish the received terrestrial cloud broadcast signals from each other and demodulate the distinguished signals.

In other words, under the situation where there is the same channel interference and timing and frequency sync is not guaranteed for each transmitted signal, one or more cloud broadcast signals should be demodulated.

For this, the terrestrial cloud broadcast system needs to operate in an environment where noise power is larger than the power of the broadcast signal, i.e., a negative SNR (Signal to Noise Ratio) environment. Accordingly, a need exists for an error correction code that operates even in such a negative SNR environment for terrestrial cloud broadcast.

SUMMARY

An object of the present invention is to provide an LDPC encoder and LDPC encoding method for encoding input information based on an LDPC (Low Density Parity Check).

Another object of the present invention is to provide an LDPC (Low Density Parity Check) code that operates even in a negative SNR (Signal to Noise Ratio) environment for terrestrial cloud broadcast.

Still another object of the present invention is to provide an LDPC code having better performance and lower complexity than the existing LDPC code.

According to an aspect of the present invention, a method of encoding input information based on an LDPC (Low Density Parity Check) includes receiving information and encoding the input information with an LDPC codeword using a parity check matrix, wherein the parity check matrix may have a structure obtained by combining a first parity check matrix for an LDPC code having a higher code rate than a reference value with a second parity check matrix for an LDPC code having a lower code rate than the reference value.

In an embodiment, the parity check matrix may include a zero matrix, an identity matrix, and a dual diagonal matrix.

In another embodiment, the encoded LDPC codeword may include a systematic part corresponding to the input information, a first parity part corresponding to the dual diagonal matrix, and a second parity part corresponding to the identity matrix.

In another embodiment, said encoding may comprise calculating the first parity part using the first parity check matrix and the input information and calculating the second parity part using the second parity check matrix based on the input information and the calculated first parity part.

In another embodiment, an element matrix of the dual diagonal matrix may constitute a dual diagonal line and is an identity matrix, and a remaining element matrix of the dual diagonal matrix may be a zero matrix.

In another embodiment, an element matrix of the dual diagonal matrix constituting a dual diagonal line may be continuous to an element matrix that constitutes a diagonal line of the identity matrix.

In another embodiment, the method may further comprise, before said encoding, determining a code rate of the LDPC code and determining a size of the dual diagonal matrix according to the determined code rate.

In another embodiment, the parity check matrix may include a zero matrix and a circulant permutation matrix.

According to another aspect of the present invention, an LDPC encoder comprises an input unit receiving information and an encoding unit encoding the input information with an LDPC codeword using a parity check matrix, wherein the parity check matrix may have a structure obtained by combining a first parity check matrix for an LDPC code having a higher code rate than a reference value with a second parity check matrix for an LDPC code having a lower code rate than the reference value.

According to still another aspect of the present invention, a method of decoding an LDPC code by an LDPC (Low Density Parity Check) decoder comprises receiving an LDPC codeword encoded by a parity check matrix and decoding the received LDPC codeword using the parity check matrix, wherein the parity check matrix may have a structure obtained by combining a first parity check matrix for an LDPC code having a higher code rate than a reference value with a second parity check matrix for an LDPC code having a lower code rate than the reference value.

According to yet still another aspect of the present invention, an LDPC decoder comprises a receiving unit receiving an LDPC (Low Density Parity Check) codeword encoded by a parity check matrix and a decoding unit decoding the received LDPC codeword using the parity check matrix, wherein the parity check matrix may have a structure obtained by combining a first parity check matrix for an LDPC code having a higher code rate than a reference value with a second parity check matrix for an LDPC code having a lower code rate than the reference value.

In a terrestrial cloud broadcast system, a LDPC (Low Density Parity Check) may be provided that operates in a negative SNR (Signal to Noise Ratio) environment.

This code provides better performance and lower complexity than those provided by the LDPC code used in the DVB-T2 (Digital Video Broadcasting-Terrestrial version 2) and DVB-S2 (Digital Video Broadcasting-Satellite-Second Generation) system.

BRIEF DESCRIPTION OF THE DRAWINGS

The embodiments of the present invention will become readily apparent by reference to the following detailed description when considered in conjunction with the accompanying drawings wherein:

FIG. 1 is a view illustrating a PCM (Parity Check Matrix) structure of a QC-LDPC (Quasi-Cyclic Low Density Parity Check) code used in a DVB (Digital Video Broadcasting) system;

FIG. 2 is a flowchart illustrating a method of encoding input information with an LDPC code that operates in a negative SNR (Signal to Noise Ratio) environment according to an embodiment of the present invention;

FIG. 3 is a view illustrating a PCM structure of an LDPC code according to an embodiment of the present invention;

FIG. 4 is a view illustrating the PCM structure of a QC-LDPC code for terrestrial cloud broadcast, wherein the length of the codeword is 8192;

FIG. 5 is an expanded view of the PCM structure shown in FIG. 4;

FIG. 6 is a view illustrating the PCM structure of a QC-LDPC code for terrestrial cloud broadcast, wherein the length of the codeword is 16384;

FIG. 7 is a view illustrating the PCM structure of a QC-LDPC code for terrestrial cloud broadcast, wherein the length of the codeword is 32768;

FIG. 8 is a view illustrating the PCM structure of a QC-LDPC code for terrestrial cloud broadcast, wherein the length of the codeword is 65536.

FIG. 9 illustrates an example of the PCM structure of an LDPC code according to code rates according to an embodiment of the present invention;

FIG. 10 is a graph illustrating the performance of LDPC codes according to an embodiment of the present invention;

FIG. 11 is a flowchart illustrating a method of decoding an LDPC codeword according to an embodiment of the present invention; and

FIG. 12 is a block diagram illustrating an LDPC encoder and an LDPC decoder according to an embodiment of the present invention.

DESCRIPTION OF THE EMBODIMENTS

Embodiments of the present invention will be described in detail with reference to the accompanying drawings to be worked on by those of ordinary skill in the art. However, the present invention may be embodied in various ways and is not limited thereto. What is irrelevant to the present invention has been omitted from the drawings, and similar denotations have been assigned to similar components throughout the specification.

As used herein, when an element “includes” or “comprises” another element, the element, unless stated otherwise, may further include or comprise the other element, but not excluding the other element. Further, as used herein, the term “unit” or “part” means a basis for processing at least one function or operation, which may be implemented in hardware or software or in a combination of software and hardware.

FIG. 1 is a view illustrating a PCM (Parity Check Matrix) structure of a QC-LDPC (Quasi-Cyclic Low Density Parity Check) code used in a DVB (Digital Video Broadcasting) system.

In general, the LDPC code is known as an error correction code that is closest to the Shannon limit in an AWGN (Additive White Gaussian Noise) channel and provides asymptotically better performance than that provided by the Turbo code while enabling parallelizable decoding.

Such an LDPC code is defined by a low-density PCM (Parity Check Matrix) that is randomly generated. However, the randomly generated LDPC code needs a lot of memory to store the PCM and requires a long time to access the memory.

To address such a problem with the memory, the QC-LDPC (Quasi-Cyclic LDPC) code has been suggested, and the QC-LDPC code constituted of Zero matrixes or CPMs (Circulant Permutation Matrixes) is defined by PCM(H) as shown in Equation 1:

$\begin{matrix} {{H = \begin{bmatrix} P^{a_{11}} & P^{a_{12}} & \ldots & P^{a_{1\; n}} \\ P^{a_{21}} & P^{a_{22}} & \ldots & P^{a_{2\; n}} \\ \vdots & \vdots & \ddots & \vdots \\ P^{a_{m\; 1}} & P^{a_{m\; 2}} & \ldots & P^{a_{mn}} \end{bmatrix}},{{{for}\mspace{14mu} a_{ij}} \in \left\{ {0,1,\ldots \mspace{14mu},{L - 1},\infty} \right\}}} & \left\lbrack {{Equation}\mspace{14mu} 1} \right\rbrack \end{matrix}$

Here, P is a CPM having a size of L×L and is defined as in Equation 2:

$\begin{matrix} {P_{L \times L} = {\begin{bmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & 1 \\ 1 & 0 & 0 & \ldots & 0 \end{bmatrix}.}} & \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack \end{matrix}$

Further, P^(i) is a matrix obtained by right-shifting, i(0≦i<L) times, an identity matrix I( ^(Δ) P⁰) having a size of L×L, and P^(∞) is the zero matrix having a size of L×L. Accordingly, in the case of the QC-LDPC code, only exponent i may be stored in order to store P^(i), and thus, the memory required for storing PCM is reduced to 1/L.

As an example, the QC-LDPC code used in the DVB system, as shown in FIG. 1, is constituted of I matrixes and P matrixes. The I matrix is a matrix having a size of (N−K)×K, and the P matrix is a dual diagonal matrix having a size of (N−K)×(N−K). Here, N is the length of codeword, and K is the length of input information.

FIG. 2 is a flowchart illustrating a method of encoding input information with an LDPC code that operates in a negative SNR (Signal to Noise Ratio) environment according to an embodiment of the present invention, and FIG. 3 is a view illustrating a PCM structure of an LDPC code according to an embodiment of the present invention.

Referring first to FIG. 2, the LDPC encoder according to the present invention, if information to be encoded is entered (210) and a code rate is determined (220), determines the size of a dual diagonal matrix that is variably included in the parity check matrix in accordance with the determined code rate (230). At this time, although in FIG. 2 the code rate is determined after the information to be encoded is entered, the code rate, as necessary, may be previously determined before the information is input or may be determined when the information is input.

Thereafter, the LDPC encoder, as shown in FIG. 3, may encode the input information with an LDPC codeword using a determined parity check matrix. Here, the parity check matrix may have a structure in which a first parity check matrix (PCM[A B]) for a LDPC code having a higher code rate a reference value (e.g., 0.5) and a second parity check matrix (PCM[C D]) for an LDPC code having a lower code rate lower than the reference value are combined with each other.

As an example, referring to FIG. 3, the parity check matrix according to the present invention may include a dual diagonal matrix B, an identity matrix D, and a zero matrix. Here, in the dual diagonal matrix B, an element matrix constituting the dual diagonal line may be an identity matrix, and the remaining element matrixes may be zero matrixes. The element matrix constituting the dual diagonal line of the dual diagonal matrix B may be continuous to the element matrix constituting the diagonal line of the identity matrix D.

In FIG. 3, N is the length of the codeword, K is the length of the information, and g is a value that varies depending on the code rate. Matrix A and matrix C have sizes of g×K and (N−K−g)×(K+g), respectively, and may be constituted of circulant permutation matrixes and zero matrixes having a size of L×L. Further, matrix Z is a zero matrix having a size of g×(N−K−g), and matrix B is a dual diagonal matrix having a size of g×g. Matrix B may be expressed as in Equation 3:

$\begin{matrix} {B_{g \times g} = {\begin{bmatrix} I_{L \times L} & 0 & 0 & \ldots & 0 & 0 & 0 \\ I_{L \times L} & I_{L \times L} & 0 & \ldots & 0 & 0 & 0 \\ 0 & I_{L \times L} & I_{L \times L} & \ldots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \ldots & I_{L \times L} & I_{L \times L} & 0 \\ 0 & 0 & 0 & \ldots & 0 & I_{L \times L} & I_{L \times L} \end{bmatrix}.}} & \left\lbrack {{Equation}\mspace{14mu} 3} \right\rbrack \end{matrix}$

Here, is an element matrix and is an identity matrix having a size of L×L.

The LDPC codeword that is encoded through the parity check matrix as shown in FIG. 3 may include the systematic part of (N−K)×K corresponding to the input information, a first parity part corresponding to the dual diagonal matrix B, and a second parity part corresponding to the identity matrix D.

The LDPC encoder according to the present invention may calculate the first parity part using the input information and the first parity check matrix (PCM[A B]) and may calculate the second parity part using the second parity check matrix (PCM[C D]) based on the input information and the calculated first parity part.

Specifically, the LDPC encoder according to the present invention may encode input information using Equation 4 as follows:

Hc ^(T)=0  [Equation 4]

Here, H is the parity check matrix, and c is the LDPC codeword.

Meanwhile, Equation 4 may be split as shown in Equation 5:

As ^(T) +Bp ₁ ^(T)=0

C(s,p ₁)^(T) +Dp ₂ ^(T)=0  [Equation 5]

Here, s is the systematic part, P₁ is the first parity part, and P₂ is the second parity part.

Since B is a dual diagonal matrix, the encoding process for calculating the first parity part P₁ may be performed by an accumulator of a block type. Further, since D is an identity matrix, the second parity part P₂ may be simply calculated by p₂ ^(T)=C(s, p₁)^(T). As such, the LDPC encoder according to the present invention has an efficient linear time encoding algorithm and thus its complexity is reduced.

As an example, N, K, and g of LDPC codes having a code rate 0.25, which have codeword lengths of 8192, 16384, 32768, and 65536, respectively, are given as in Table 1:

TABLE 1 N K g 8192 2048 160 16384 4096 320 32768 8192 640 65536 16384 1280

An exemplary method of expressing the PCM of the QC-LDPC code designed as in Table 1 is now described.

Assume that a QC-LDPC code having a code rate of 4/, N=28 and K=16, and that is constituted of a 4×4 CPM has a PCM as shown in Equation 6:

$\begin{matrix} {{H = \begin{bmatrix} P^{2} & P^{3} & P^{\infty} & P^{\infty} & P^{0} & P^{\infty} & P^{\infty} \\ P^{\infty} & P^{1} & P^{\infty} & P^{2} & P^{\infty} & P^{0} & P^{\infty} \\ P^{2} & P^{3} & P^{0} & P^{\infty} & P^{\infty} & P^{\infty} & P^{0} \end{bmatrix}}\begin{pmatrix} {= \begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}} \\ {{P^{0} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}},{P^{1} = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix}},{P^{2} = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}},{P^{3} = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}},{P^{\infty} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}}} \end{pmatrix}} & \left\lbrack {{Equation}\mspace{14mu} 6} \right\rbrack \end{matrix}$

The PCM may be represented as follows.

7 3 Matrix size (# of columns = 7, # of rows = 3) 2 3 1 1 1 1 1 Column weight distribution 0^(th) column weight = 2, 1^(st) column weight = 3, 2^(nd) column weight = 1, 3^(rd) column weight = 1, 4^(th) column weight = 1, 5^(th) column weight = 1, 6^(th) column weight = 1. 3 3 4 Row weight distribution 0^(th) row weight = 3, 1^(st) row weight = 3, 2^(nd) row weight = 4. Location of non-zero matrix at each row 0 1 4 Non-zero matrix location of 0^(th) row, row weight = 3 1 3 5 Non-zero matrix location of 1^(st) row, row weight = 3 0 1 2 6 Non-zero matrix location of 2^(nd) row, row weight = 4 Exponent value of non-zero matrix at each row 2 3 0 Exponent value of non-zero matrix of 0^(th) row 1 2 0 Exponent value of non-zero matrix of 1^(st) row 2 3 0 0 Exponent value of non-zero matrix of 2^(nd) row

The PCM of the QC-LDPC code with a code rate of 0.25, which operates in a negative SNR environment for a terrestrial cloud broadcast system according to the present invention may be represented in such way, thus resulting in the following embodiments:

[Embodiment 1] 256 192 => Matrix Size (# of columns = 256, # of rows = 192) Column weight distribution Ex) 0-th column weight = 14, 1-st column weight = 14, ..., 63-th column weight = 13, 64-th column weight = 17, ..., 68-th column weight = 17,   69-th column weight = 1, ..., 255-th column weight = 1. 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 17 17 17 17 17 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Row weight distribution Ex) 0-th row weight = 40, 1-st row weight = 41, ..., 4-th row weight = 39, 5-th row weight = 5, ..., 191-th row weight = 5. 40 41 41 40 39 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 Location of non-zero matrix (circulant permutation matrix) at each row 0 1 4 5 6 7 8 9 12 13 14 15 16 17 20 21 23 25 27 28 29 32 33 36 38 39 41 43 47 49 51 52 53 57 58 60 61 62 63 64     ==> Non-zero matrix location of 0-th row, row weight = 40 2 3 4 5 9 10 11 12 13 14 16 18 19 20 21 22 23 25 26 27 28 30 35 36 39 40 41 42 43 44 45 46 50 51 55 56 57 58 59 64 65 ==> Non-zero matrix location of 1-st row, row weight = 41 0 1 2 4 5 6 7 8 10 12 14 15 16 17 21 22 23 24 26 30 31 32 33 34 37 38 39 40 44 45 47 48 49 51 53 54 55 56 57 65 66 0 2 3 10 11 13 15 18 19 24 25 26 27 29 31 33 34 35 37 40 41 42 44 45 46 48 50 52 53 54 55 56 58 59 60 61 62 63 66 67 1 3 6 7 8 9 11 17 18 19 20 22 24 28 29 30 31 32 34 35 36 37 38 42 43 46 47 48 49 50 52 54 59 60 61 62 63 67 68     ==> Non-zero matrix location of 4-th row, row weight = 39 3 21 51 68 69 ==> Non-zero matrix location of 5-th row, row weight = 5 9 24 47 48 70 5 19 43 61 71 11 33 47 55 72 3 31 41 65 73 15 22 45 49 74 0 7 60 65 75 9 18 36 37 76 1 21 45 65 77 14 21 44 59 78 9 35 52 65 79 9 30 41 64 80 0 23 46 66 81 12 27 38 39 82 18 34 50 65 83 0 19 32 64 84 6 19 45 57 85 5 8 47 51 86 17 20 45 48 87 13 29 64 66 88 2 22 39 59 89 15 26 50 67 90 8 26 52 57 91 5 25 42 55 92 0 26 37 51 93 10 20 30 54 94 12 21 36 61 95 7 29 35 61 96 8 30 40 66 97 12 22 51 63 98 17 27 44 64 99 17 34 46 68 100 7 33 42 64 101 18 23 35 53 102 8 14 55 60 103 4 21 30 43 104 0 27 56 63 105 5 24 58 67 106 12 20 49 64 107 2 6 38 65 108 6 30 53 60 109 15 28 47 65 110 16 22 26 32 111 1 23 42 64 112 12 19 55 65 113 4 23 40 49 114 9 28 31 53 115 8 23 32 65 116 16 24 65 67 117 11 24 61 67 118 16 29 31 68 119 1 40 59 61 120 8 20 38 53 121 11 20 65 68 122 17 25 53 66 123 9 25 50 51 124 9 38 40 58 125 6 25 56 59 126 18 30 59 66 127 6 12 31 68 128 13 33 45 63 129 10 25 43 58 130 3 32 48 63 131 10 28 66 68 132 4 19 31 42 133 14 33 38 43 134 13 20 28 52 135 5 34 37 48 136 6 39 47 52 137 7 27 31 43 138 1 25 47 64 139 5 32 35 36 140 14 24 31 64 141 18 32 52 54 142 3 10 50 52 143 11 26 43 62 144 18 31 47 68 145 9 42 46 49 146 4 32 51 65 147 14 34 56 66 148 0 5 49 50 149 17 37 55 68 150 1 22 31 66 151 1 27 30 48 152 0 22 30 61 153 3 38 42 45 154 14 35 51 57 155 14 29 42 58 156 2 4 54 68 157 6 11 28 41 158 16 33 48 62 159 1 39 50 62 160 7 20 57 62 161 4 27 46 52 162 16 20 46 60 163 10 27 47 60 164 15 23 52 60 165 14 18 39 41 166 10 19 59 63 167 14 28 45 67 168 12 28 62 67 169 7 32 37 46 170 1 37 41 67 171 8 39 48 49 172 7 18 63 67 173 10 26 39 53 174 13 21 46 57 175 17 33 39 65 176 11 18 45 56 177 0 40 45 52 178 15 19 44 51 179 4 17 28 61 180 8 36 44 58 181 11 25 54 63 182 4 24 50 55 183 2 34 61 66 184 5 29 59 62 185 2 43 52 63 186 2 30 44 62 187 14 22 50 68 188 18 27 49 62 189 4 37 38 60 190 13 26 38 64 191 12 33 67 68 192 17 18 42 57 193 3 29 67 68 194 16 28 39 64 195 13 24 54 67 196 0 36 43 53 197 15 21 32 34 198 8 16 27 59 199 7 19 58 68 200 2 33 50 60 201 16 21 40 63 202 2 42 47 67 203 1 19 26 54 204 13 22 53 55 205 17 29 40 50 206 15 24 30 35 207 14 19 30 37 208 17 26 41 49 209 1 29 43 51 210 16 19 56 67 211 11 21 35 58 212 15 27 40 53 213 13 23 25 48 214 6 36 46 67 215 10 23 55 62 216 4 20 34 67 217 12 24 57 60 218 3 36 40 64 219 7 21 54 66 220 7 28 48 66 221 7 24 25 68 222 15 20 41 56 223 10 22 38 48 224 12 29 32 41 225 16 23 47 61 226 5 33 44 66 227 6 13 37 49 228 4 36 41 59 229 10 33 36 56 230 17 22 35 43 231 0 2 3 68 232 15 29 36 54 233 13 27 35 42 234 5 21 53 56 235 10 34 57 64 236 11 22 64 66 237 0 41 57 58 238 9 26 56 66 239 13 31 34 58 240 8 25 46 65 241 3 35 59 60 242 5 28 54 57 243 1 2 9 55 244 6 32 55 61 245 2 15 37 64 246 9 20 44 65 247 11 23 38 51 248 12 26 34 40 249 8 31 56 62 250 11 29 49 66 251 6 23 44 63 252 3 24 44 46 253 16 25 44 45 254 ==> Non-zero matrix location of 190-th row, row weight = 5 3 39 54 58 255 ==> Non-zero matrix location of 191-th row, row weight = 5 Exponent value of non-zero matrix at each row 12 22 31 15 21 31 22 3 12 28 13 13 27 13 11 23 29 13 13 28 26 28 0 6 1 20 17 31 28 8 25 23 16 4 29 14 7 10 23 0     ==> Exponent value of non-zero matrix of 0-th row 14 23 1 2 9 0 9 9 12 6 4 0 7 23 6 0 5 3 23 14 22 28 15 13 1 7 3 18 30 24 10 15 31 3 13 21 8 27 8 0 0         ==> Exponent value of non-zero matrix of 1-st row 20 9 29 9 19 22 16 5 16 18 9 25 6 5 20 4 2 29 14 1 15 21 31 2 28 3 20 28 25 17 23 3 15 13 25 4 25 9 26 0 0 30 20 21 23 24 1 21 27 25 17 15 28 10 27 8 12 18 25 4 8 6 25 12 12 9 6 8 29 15 2 17 3 24 21 4 16 26 14 0 0 2 30 7 26 3 28 3 24 1 27 2 27 3 0 8 5 28 15 9 15 12 23 6 23 28 17 20 27 4 5 22 3 25 14 28 11 6 0 0          ==> Exponent value of non-zero matrix of 4-th row 11 0 4 0 0 ==> Exponent value of non-zero matrix of 5-th row 5 16 12 23 0 24 4 11 3 0 0 14 1 10 0 26 17 7 0 0 4 30 14 19 0 28 26 23 0 0 6 2 31 9 0 5 14 27 0 0 30 14 8 18 0 17 19 30 0 0 17 18 10 0 0 25 11 23 0 0 10 29 7 11 0 29 10 30 0 0 30 23 10 0 0 8 20 16 3 0 25 26 26 28 0 20 20 24 1 0 23 18 0 0 0 0 12 31 22 0 22 4 26 0 0 30 26 30 26 0 30 29 8 19 0 14 30 27 22 0 4 19 28 13 0 3 9 9 28 0 20 4 23 6 0 20 15 0 0 0 27 22 14 3 0 18 10 26 0 0 1 26 25 0 0 11 8 16 6 0 4 15 9 30 0 5 0 13 26 0 20 21 16 24 0 18 5 18 6 0 1 27 29 0 0 24 27 3 0 0 6 7 3 23 0 19 24 17 5 0 28 19 17 0 0 10 17 3 23 0 31 28 2 0 0 6 23 17 5 0 15 0 19 0 0 31 0 1 14 0 20 18 3 0 0 15 13 21 0 0 31 14 18 0 0 1 10 18 0 0 7 9 3 7 0 14 15 28 31 0 12 3 0 0 0 25 20 0 11 0 20 29 2 31 0 22 28 17 15 0 6 10 16 28 0 17 24 17 0 0 12 18 0 0 0 9 21 26 31 0 20 13 21 31 0 23 2 11 4 0 17 10 20 0 0 4 23 28 0 0 1 1 28 6 0 4 9 12 13 0 22 27 15 29 0 28 7 23 13 0 0 23 8 22 0 29 4 13 0 0 8 4 22 11 0 28 7 21 0 0 7 9 19 23 0 13 27 4 23 0 14 24 23 7 0 8 0 9 12 0 10 14 2 1 0 8 27 21 16 0 11 20 24 0 0 4 31 17 13 0 20 3 13 24 0 9 22 12 22 0 8 5 22 4 0 27 14 0 4 0 3 28 27 3 0 5 12 13 23 0 27 25 26 18 0 8 30 2 24 0 14 16 2 31 0 9 30 17 26 0 9 25 31 31 0 2 29 18 11 0 10 4 19 27 0 23 6 17 15 0 25 29 28 29 0 15 14 22 2 0 19 23 19 6 0 17 28 8 15 0 20 19 26 0 0 1 6 10 0 0 9 24 27 20 0 17 0 16 17 0 31 12 17 1 0 4 0 27 0 0 9 16 15 22 0 30 15 19 5 0 0 28 11 0 0 3 21 2 2 0 27 16 31 24 0 15 20 10 2 0 28 15 27 29 0 25 30 18 16 0 30 26 31 21 0 5 24 2 31 0 31 16 10 0 0 9 25 4 8 0 9 6 9 15 0 11 11 13 11 0 6 18 1 0 0 13 12 23 13 0 25 18 31 17 0 26 20 5 0 0 12 4 6 0 0 3 8 28 19 0 14 18 5 0 0 10 16 31 0 0 19 24 27 31 0 1 25 3 0 0 23 29 10 8 0 2 2 5 20 0 16 14 6 12 0 21 22 20 9 0 12 21 25 2 0 9 8 25 0 0 5 27 13 23 0 21 14 12 9 0 27 0 25 4 0 8 25 8 3 0 30 7 14 2 0 28 11 11 1 0 26 27 31 11 0 12 12 24 0 0 28 10 5 0 0 20 22 17 30 0 15 13 19 2 0 13 2 24 0 0 0 0 0 18 0 28 19 28 5 0 28 29 7 24 0 25 2 8 16 0 9 14 16 0 0 31 27 2 0 0 1 8 22 0 0 9 24 6 13 0 7 31 23 29 0 22 12 3 19 0 10 31 25 14 0 7 5 20 0 0 19 16 30 24 0 11 31 4 27 0 0 0 30 3 0 12 1 27 17 0 14 8 28 0 0 1 27 9 13 0 19 27 24 5 0 0 25 28 27 0 9 31 5 0 0 27 28 30 0 0 17 18 16 23 0 4 8 21 0 0 18 19 30 24 0 23 7 28 0 0 29 2 10 7 0 5 24 11 4 0 16 18 7 12 0 1 7 24 3 0 2 21 25 0 0 5 8 22 0 0 7 1 23 19 0 12 13 13 0 0 23 13 11 3 0 16 29 23 0 0 7 7 21 11 0 22 30 31 0 0 23 30 15 26 0 ==> Exponent value of non-zero matrix of 190-th row 12 14 22 26 0 ==> Exponent value of non-zero matrix of 191-th row

Embodiment 2 512 384 => Matrix Size (# of columns = 512, # of rows = 384) Column weight distribution Ex) 0-th column weight = 14, 1-st column weight = 14, . . . , 127-th column weight = 13, 128- th column weight = 17, . . . , 137-th column weight = 17, 138-th column weight = 1, . . . , 511-th column weight = 1. 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 17 17 17 17 17 17 17 17 17 17 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Row weight distribution Ex) 0-th row weight = 41, 1-st row weight = 37, . . . , 9-th row weight = 41, 10-th row weight = 5, . . . , 383-th row weight = 5. 41 37 41 41 39 41 40 41 41 41 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 Location of non-zero matrix (circulant permutation matrix) at each row 2 3 5 11 16 18 19 22 25 26 30 32 37 40 42 44 48 50 54 59 60 64 70 71 73 75 79 82 84 92 96 102 103 104 109 112 114 116 117 120 128 ==> Non-zero matrix location of 0-th row, row weight = 41 4 8 15 20 27 29 32 36 39 43 56 57 60 62 65 66 68 69 71 76 85 89 90 94 95 98 102 106 108 114 115 120 121 124 127 128 129 ==> Non-zero matrix location of 1-st row, row weight = 37 3 10 13 20 22 28 29 32 33 34 37 40 45 49 53 55 56 57 58 60 64 68 75 77 80 86 89 91 94 102 107 108 110 111 116 119 121 125 126 129 130 1 6 9 13 15 18 21 22 24 31 33 36 38 41 48 49 51 52 55 59 62 72 78 80 83 86 87 88 93 94 97 99 100 101 103 104 105 106 107 130 131 2 5 12 13 23 25 26 27 38 43 45 47 48 50 52 53 57 58 63 64 67 69 72 75 76 83 90 92 98 99 111 113 115 117 119 121 127 131 132 0 7 8 10 11 12 14 18 19 31 34 35 37 40 42 46 54 67 68 71 74 76 81 84 85 87 88 91 100 103 109 110 114 117 118 119 122 123 125 132 133 0 4 5 6 9 21 24 29 30 34 35 41 42 44 46 51 52 61 65 66 70 77 78 79 81 83 84 85 89 91 95 98 101 104 112 122 123 127 133 134 1 6 7 10 12 16 17 19 23 24 25 28 30 36 39 44 49 56 58 59 61 66 70 73 77 86 90 93 96 97 101 106 107 110 113 115 116 120 122 134 135 1 4 9 14 15 16 17 20 27 28 31 33 38 39 41 45 47 50 53 61 62 63 67 69 74 79 80 82 87 88 92 95 99 105 109 111 118 124 126 135 136 0 2 3 7 8 11 14 17 21 23 26 35 43 46 47 51 54 55 63 65 72 73 74 78 81 82 93 96 97 100 105 108 112 113 118 123 124 125 126 136 137 ==> Non-zero matrix location of 9-th row, row weight = 41 35 39 111 134 138 ==> Non-zero matrix location of 10-th row, row weight = 5 37 40 63 137 139 10 55 56 77 140 5 59 91 93 141 5 6 99 114 142 35 50 87 129 143 29 42 74 132 144 8 45 66 98 145 9 63 81 134 146 18 20 57 124 147 33 52 89 120 148 15 48 107 108 149 25 47 69 90 150 19 68 83 110 151 22 42 83 99 152 34 43 74 82 153 3 4 58 112 154 32 39 120 125 155 26 42 127 130 156 24 57 81 119 157 9 70 124 135 158 14 60 108 117 159 3 69 89 117 160 23 46 82 104 161 6 7 27 37 162 10 41 78 83 163 24 67 105 128 164 20 59 85 123 165 33 50 98 116 166 4 57 87 133 167 0 33 61 113 168 21 41 125 128 169 12 38 57 113 170 32 50 105 130 171 24 50 78 131 172 27 70 80 106 173 1 39 104 119 174 2 77 102 109 175 28 57 97 120 176 33 57 108 112 177 16 51 117 129 178 17 49 70 84 179 2 50 86 120 180 11 61 101 105 181 35 44 56 127 182 14 40 119 122 183 0 15 63 135 184 3 39 101 128 185 31 58 83 121 186 22 59 63 132 187 9 62 110 129 188 32 37 84 133 189 30 43 101 125 190 3 47 76 113 191 3 63 82 137 192 27 72 92 111 193 32 67 90 99 194 22 67 80 87 195 5 44 72 75 196 25 74 118 130 197 27 59 121 130 198 2 6 82 88 199 13 39 108 124 200 0 55 124 136 201 11 63 75 90 202 6 56 91 118 203 9 68 85 125 204 8 9 106 133 205 29 49 99 110 206 17 54 80 91 207 14 42 52 134 208 12 61 112 136 209 3 12 106 137 210 33 53 67 103 211 26 63 96 111 212 12 40 51 85 213 12 56 94 119 214 35 40 118 132 215 12 79 110 133 216 19 48 69 125 217 25 52 114 124 218 32 48 59 86 219 24 65 116 130 220 1 56 90 123 221 7 71 96 104 222 14 69 78 104 223 37 44 88 95 224 0 62 119 132 225 13 62 131 136 226 29 60 64 118 227 2 84 85 118 228 30 41 94 111 229 7 45 113 127 230 8 50 94 95 231 15 45 77 121 232 32 80 82 132 233 36 38 88 107 234 26 71 113 132 235 20 69 111 128 236 17 61 75 97 237 30 68 113 129 238 9 53 92 128 239 12 68 72 128 240 32 42 89 122 241 20 42 81 136 242 12 52 82 130 243 36 54 75 95 244 7 58 118 120 245 28 54 86 103 246 18 40 100 116 247 1 66 86 97 248 34 40 80 89 249 6 76 89 132 250 17 63 69 99 251 19 44 120 133 252 8 38 85 132 253 36 55 103 109 254 12 53 60 135 255 9 39 100 127 256 21 55 123 129 257 20 25 77 133 258 20 46 62 68 259 32 72 76 107 260 7 41 101 131 261 26 45 88 104 262 30 38 90 137 263 18 53 121 125 264 25 70 119 123 265 15 24 91 136 266 24 56 71 121 267 10 81 88 91 268 5 40 105 129 269 16 41 77 84 270 7 79 102 116 271 27 50 99 109 272 28 52 91 128 273 14 56 98 110 274 20 61 93 99 275 11 67 82 119 276 10 69 79 119 277 33 59 79 129 278 34 65 77 97 279 4 65 107 134 280 27 51 91 120 281 18 36 98 129 282 11 13 58 98 283 24 72 93 94 284 1 48 96 110 285 21 37 126 131 286 11 49 126 135 287 8 60 112 137 288 5 9 57 136 289 23 45 123 131 290 6 49 86 98 291 22 74 94 113 292 17 38 129 133 293 27 62 75 124 294 19 45 96 132 295 23 55 130 132 296 6 48 81 120 297 21 51 90 130 298 35 49 65 81 299 31 85 108 129 300 0 28 51 116 301 25 45 65 100 302 35 43 115 137 303 4 68 115 117 304 9 23 43 89 305 16 44 64 79 306 10 61 86 128 307 19 47 92 129 308 2 44 100 125 309 36 61 69 80 310 3 50 74 121 311 1 42 103 129 312 31 56 92 114 313 16 46 76 128 314 19 46 89 136 315 27 60 100 128 316 16 53 109 136 317 21 62 71 135 318 2 43 69 129 319 34 56 64 135 320 22 38 71 137 321 22 58 91 133 322 30 44 80 99 323 36 74 104 117 324 31 65 76 93 325 14 59 124 127 326 37 61 122 135 327 7 67 75 133 328 24 39 64 132 329 13 14 82 115 330 11 64 92 131 331 33 46 107 135 332 3 44 123 134 333 17 43 60 131 334 24 41 52 115 335 13 81 111 114 336 11 38 108 111 337 13 57 117 135 338 8 48 105 135 339 23 58 92 99 340 17 47 102 103 341 4 62 104 121 342 36 52 64 97 343 17 51 79 134 344 1 57 95 101 345 16 32 49 115 346 4 6 26 106 347 25 37 94 103 348 1 7 93 128 349 30 46 70 132 350 23 64 84 94 351 36 46 102 121 352 2 53 122 130 353 22 70 112 114 354 0 66 83 87 355 31 64 89 109 356 27 54 65 69 357 13 44 54 90 358 15 75 87 103 359 15 68 116 126 360 10 45 87 136 361 36 48 77 78 362 5 71 111 117 363 12 46 78 118 364 2 45 62 133 365 3 42 90 135 366 10 18 50 110 367 13 41 95 133 368 34 70 95 115 369 28 43 96 106 370 2 51 73 94 371 30 48 51 102 372 21 57 80 83 373 20 76 98 135 374 11 74 85 102 375 20 34 106 132 376 29 63 113 136 377 31 37 53 79 378 18 49 87 134 379 26 53 55 114 380 29 66 95 134 381 4 52 127 131 382 26 54 83 115 383 10 72 112 121 384 19 40 86 131 385 31 48 62 134 386 31 82 87 106 387 5 11 52 107 388 8 44 102 130 389 10 29 131 133 390 27 42 87 97 391 12 45 75 86 392 8 40 55 88 393 14 15 73 74 394 6 66 101 126 395 4 43 105 114 396 15 70 93 127 397 6 64 112 134 398 17 52 88 122 399 35 62 80 114 400 13 53 78 129 401 0 49 128 130 402 34 83 104 130 403 3 49 108 132 404 18 71 82 109 405 21 43 72 110 406 22 54 128 135 407 1 65 85 92 408 26 39 75 85 409 23 49 73 78 410 35 47 100 136 411 31 41 81 124 412 7 78 103 130 413 33 39 93 117 414 28 71 99 134 415 5 50 83 84 416 29 38 65 124 417 16 71 95 107 418 14 67 84 137 419 35 77 130 135 420 35 73 117 137 421 23 54 119 120 422 22 41 60 107 423 19 60 93 109 424 26 70 79 101 425 2 57 92 134 426 32 60 123 137 427 25 71 79 89 428 21 42 78 109 429 29 47 116 123 430 28 46 100 111 431 4 73 81 109 432 14 16 61 83 433 1 40 72 126 434 34 48 88 118 435 30 65 73 131 436 18 66 123 136 437 13 70 76 105 438 24 47 59 120 439 22 35 66 68 440 7 43 98 121 441 18 75 112 126 442 23 59 97 122 443 1 46 122 133 444 15 58 80 90 445 31 74 86 116 446 30 63 118 119 447 17 31 94 118 448 25 60 129 136 449 16 66 130 133 450 25 58 72 131 451 8 49 104 109 452 20 55 97 116 453 5 46 131 134 454 13 66 89 116 455 29 58 77 122 456 30 53 91 104 457 33 58 96 101 458 26 74 90 93 459 24 37 106 110 460 18 64 73 122 461 2 22 61 115 462 15 54 84 101 463 23 51 76 114 464 9 47 107 137 465 27 38 96 136 466 29 54 106 137 467 28 78 85 101 468 8 52 76 92 469 18 58 88 137 470 9 56 86 113 471 26 41 92 98 472 10 39 51 105 473 16 47 93 111 474 28 67 81 107 475 6 73 102 128 476 15 53 88 123 477 25 51 87 122 478 0 19 106 117 479 5 60 103 137 480 34 38 105 126 481 36 45 105 108 482 11 48 100 112 483 28 37 68 127 484 33 56 125 132 485 16 43 103 127 486 8 61 125 134 487 4 10 84 96 488 20 44 96 126 489 21 54 96 124 490 19 21 84 91 491 0 41 102 114 492 19 38 73 98 493 0 42 95 131 494 36 63 72 79 495 0 64 115 126 496 28 47 112 135 497 34 66 110 113 498 29 62 67 94 499 7 47 55 137 500 14 30 97 100 501 34 39 102 131 502 3 67 77 136 503 21 59 76 100 504 11 55 68 134 505 5 65 125 133 506 32 55 73 126 507 33 40 97 127 508 4 59 95 108 509 1 17 50 115 510 ==> Non-zero matrix location of 382-th row, row weight = 5 23 37 108 128 511 ==> Non-zero matrix location of 383-th row, row weight = 5 Exponent value of non-zero matrix at each row 30 7 20 12 29 17 3 31 21 7 11 2 15 24 6 15 16 24 1 8 20 4 10 22 7 11 6 26 30 21 26 19 28 18 11 20 29 19 2 30 0 ==> Exponent value of non-zero matrix of 0-th row 28 25 17 9 12 27 6 21 19 27 26 15 26 10 20 5 25 22 2 4 24 19 23 1 16 16 21 12 5 14 8 11 19 18 19 0 0 ==> Exponent value of non-zero matrix of 1- st row 6 22 24 15 4 9 20 16 14 12 18 11 19 15 20 21 6 6 0 14 2 30 27 30 29 2 11 25 29 29 12 26 14 6 0 21 4 1 3 0 0 14 28 26 15 1 5 8 21 17 23 23 13 5 0 24 11 17 16 25 19 23 25 19 13 13 17 11 3 0 26 11 4 25 22 1 10 16 13 30 0 0 10 30 5 4 31 6 11 5 11 23 11 22 11 27 13 1 29 20 23 3 19 25 15 31 16 27 1 7 11 12 12 22 31 2 20 30 4 0 0 25 15 19 24 23 24 27 25 0 26 28 24 24 8 9 14 25 14 16 16 11 9 27 19 8 20 25 19 19 2 13 1 4 17 27 25 21 20 21 0 0 17 6 12 13 8 0 22 1 21 18 22 26 26 2 2 12 28 21 29 0 18 21 22 8 29 17 1 20 23 20 16 27 15 29 6 28 19 31 0 0 12 5 15 2 5 20 21 22 17 24 29 30 24 9 16 18 6 7 31 7 28 1 26 8 20 10 30 31 14 21 13 30 14 9 4 0 16 5 9 0 0 1 28 20 4 26 21 28 16 25 21 2 18 15 31 9 9 30 21 1 0 20 29 15 30 3 27 11 4 19 27 23 30 11 12 7 24 12 20 28 0 0 5 9 16 31 1 30 16 26 26 20 24 11 10 10 14 7 29 10 26 13 30 12 28 18 8 26 10 14 12 29 13 29 28 29 0 6 5 30 31 0 0  ==> Exponent value of non-zero matrix of 9- th row 18 14 0 0 0 ==> Exponent value of non-zero matrix of 10-th row 12 18 28 0 0 9 5 7 19 0 7 10 29 28 0 8 30 27 21 0 7 0 21 0 0 9 1 13 4 0 3 5 4 6 0 10 14 9 0 0 19 24 16 3 0 7 13 23 24 0 6 24 29 16 0 9 8 5 12 0 18 6 7 24 0 19 17 14 4 0 19 25 30 15 0 26 17 26 5 0 28 2 5 12 0 22 16 7 0 0 25 19 7 18 0 8 21 22 0 0 6 11 18 8 0 18 2 29 22 0 27 12 8 27 0 2 29 7 7 0 1 17 22 23 0 22 0 12 0 0 5 4 19 19 0 0 4 22 1 0 25 24 7 24 0 14 3 2 7 0 5 9 28 11 0 31 22 10 23 0 28 29 30 1 0 26 23 18 0 0 20 21 14 27 0 16 21 26 1 0 11 5 10 17 0 27 28 28 11 0 25 3 31 21 0 11 10 12 7 0 23 1 8 31 0 9 25 31 28 0 31 10 28 16 0 28 18 5 31 0 27 5 5 22 0 13 1 0 0 0 11 26 7 21 0 29 31 5 26 0 23 5 6 5 0 7 23 25 0 0 0 5 29 1 0 3 14 26 24 0 17 26 25 17 0 24 18 6 29 0 3 25 22 7 0 30 28 4 13 0 0 19 14 17 0 9 14 25 13 0 0 12 29 7 0 8 2 29 19 0 0 7 14 17 0 4 27 23 3 0 19 5 14 21 0 16 11 7 22 0 19 23 8 7 0 26 10 1 17 0 25 17 15 11 0 21 29 11 11 0 1 28 21 31 0 23 20 17 0 0 13 12 14 19 0 11 9 11 17 0 18 31 20 7 0 23 13 18 4 0 17 13 10 29 0 9 8 27 1 0 28 1 5 0 0 13 12 31 9 0 23 6 25 22 0 13 8 8 6 0 24 25 5 7 0 25 9 21 15 0 11 11 15 3 0 25 13 16 3 0 6 2 2 10 0 8 11 27 26 0 18 1 24 13 0 15 2 17 19 0 19 20 30 14 0 25 16 17 12 0 12 30 6 3 0 1 5 3 24 0 5 3 12 24 0 25 28 18 23 0 26 29 8 0 0 24 1 29 11 0 5 11 23 0 0 20 8 31 9 0 2 22 16 7 0 20 5 23 5 0 4 14 17 0 0 10 12 8 5 0 21 13 12 10 0 8 1 8 1 0 28 17 21 0 0 2 22 14 15 0 0 15 5 30 0 18 3 21 28 0 11 22 30 5 0 8 22 9 11 0 30 6 0 21 0 27 1 22 0 0 30 29 8 16 0 21 8 31 0 0 4 20 1 0 0 4 23 23 1 0 16 23 12 0 0 11 13 0 9 0 11 21 16 28 0 26 10 25 0 0 15 19 26 8 0 19 9 14 5 0 9 29 28 0 0 15 19 13 30 0 30 27 30 0 0 4 27 19 14 0 14 30 8 8 0 11 15 17 15 0 31 6 11 3 0 5 19 5 13 0 19 29 5 0 0 0 31 9 30 0 14 1 15 4 0 24 24 11 4 0 31 15 28 0 0 2 6 20 9 0 7 17 3 6 0 3 21 24 31 0 4 13 17 23 0 25 15 7 0 0 20 17 18 23 0 25 0 22 20 0 3 11 3 3 0 13 7 7 0 0 15 16 13 30 0 24 14 20 29 0 10 24 23 25 0 26 16 17 24 0 0 24 31 0 0 13 16 30 18 0 23 5 24 31 0 4 1 20 1 0 2 30 3 24 0 29 12 10 10 0 22 21 8 0 0 6 17 21 6 0 19 13 16 0 0 26 21 26 0 0 28 17 24 30 0 28 2 22 28 0 10 25 31 24 0 17 15 25 0 0 8 0 13 11 0 22 14 29 29 0 10 13 23 24 0 11 21 29 3 0 3 29 18 0 0 4 23 9 28 0 1 26 30 9 0 18 30 11 12 0 18 27 3 23 0 8 2 31 0 0 25 0 4 27 0 2 29 11 0 0 23 30 2 3 0 27 13 23 0 0 3 20 21 20 0 7 10 13 0 0 6 29 29 0 0 19 9 18 20 0 25 12 29 0 0 21 10 8 22 0 27 12 19 2 0 22 5 15 0 0 15 27 31 5 0 15 15 8 5 0 4 4 26 5 0 0 28 0 0 0 0 25 22 0 0 3 13 8 0 0 30 23 12 11 0 21 15 3 5 0 9 17 15 11 0 19 18 7 0 0 1 29 29 7 0 18 17 25 0 0 30 31 28 4 0 15 12 19 18 0 14 21 28 7 0 19 20 27 0 0 17 25 11 0 0 28 8 0 15 0 26 0 5 31 0 16 21 15 21 0 22 8 7 22 0 16 31 6 0 0 19 30 6 8 0 7 26 15 21 0 6 4 28 25 0 26 16 29 13 0 27 4 31 0 0 11 14 28 16 0 30 7 24 8 0 15 18 20 16 0 21 13 27 0 0 19 7 3 20 0 10 12 11 29 0 18 18 25 7 0 30 29 27 2 0 7 10 5 28 0 19 2 31 4 0 8 22 29 19 0 2 14 18 0 0 14 7 6 6 0 5 9 26 27 0 31 2 19 18 0 26 24 11 0 0 21 12 28 0 0 15 14 17 23 0 7 2 0 0 0 9 6 6 1 0 13 22 11 19 0 4 10 29 1 0 9 9 16 22 0 30 0 7 6 0 29 15 12 0 0 15 14 17 28 0 22 29 22 0 0 13 18 11 16 0 16 16 29 4 0 11 13 18 0 0 18 26 29 13 0 16 24 28 0 0 10 1 14 23 0 6 15 1 29 0 8 14 25 1 0 23 18 30 0 0 23 3 8 0 0 11 29 9 9 0 16 28 4 20 0 25 31 30 0 0 27 4 0 0 0 28 4 1 22 0 23 3 2 13 0 4 8 20 19 0 12 8 13 0 0 31 15 5 17 0 24 16 29 14 0 2 11 29 1 0 6 9 9 0 0 15 31 23 5 0 7 16 19 26 0 27 1 25 0 0 26 6 0 0 0 23 5 15 0 0 13 14 15 0 0 18 6 6 7 0 27 2 8 16 0 30 18 0 0 0 23 6 2 24 0 7 23 29 27 0 31 14 0 30 0 21 2 31 0 0 19 15 11 27 0 12 20 20 0 0 24 18 5 18 0 10 26 20 0 0 22 23 13 27 0 15 15 9 29 0 13 14 26 13 0 22 13 19 24 0 11 13 0 15 0 28 3 25 0 0 13 11 6 15 0 5 13 6 25 0 27 21 6 4 0 14 18 20 3 0 19 15 3 0 0 9 25 11 0 0 12 2 19 18 0 28 8 17 11 0 21 30 4 19 0 4 10 14 11 0 27 19 19 28 0 13 7 3 24 0 6 8 25 16 0 22 19 3 16 0 24 5 7 0 0 17 28 20 0 0 1 21 0 12 0 12 3 10 22 0 5 10 21 7 0 15 15 16 15 0 30 23 18 10 0 7 24 7 3 0 6 9 9 0 0 8 6 26 1 0 25 29 28 6 0 20 2 2 27 0 29 5 0 10 0 17 13 0 21 0 15 17 22 0 0 3 3 29 0 0 9 30 13 2 0 30 9 6 31 0 4 25 27 27 0 2 26 6 29 0 25 15 31 12 0 27 25 12 31 0 15 23 4 11 0 3 30 15 23 0 0 21 24 21 0 1 7 1 20 0 5 25 18 25 0 8 23 18 9 0 21 7 2 25 0 25 20 0 0 0 8 31 23 0 0 4 26 24 0 0 15 4 4 24 0 20 9 13 24 0 16 30 1 0 0 10 16 22 0 0 23 26 15 4 0 17 12 26 29 0 20 12 29 1 0 30 6 31 27 0 21 23 16 0 0 19 21 8 31 0 0 2 3 24 0 31 30 26 12 0 27 6 10 0 0 4 14 1 11 0 16 9 21 12 0 12 9 25 14 0 19 23 1 17 0 8 29 29 0 0 16 14 4 4 0 10 23 0 0 0 9 18 0 18 0 26 24 25 11 0 15 6 10 16 0 24 28 21 14 0 3 22 23 22 0 5 4 20 11 0 27 30 17 0 0 19 8 13 4 0 27 18 4 7 0 29 28 29 0 0 22 7 6 8 0 17 25 7 31 0 1 18 26 0 0 7 2 0 8 0 24 29 6 0 0 21 22 22 0 0 31 19 19 8 0 4 1 21 0 0 14 16 10 14 0 25 4 18 14 0 24 14 30 8 0 9 22 17 12 0 4 28 4 19 0 ==> Exponent value of non-zero matrix of 382-th row 21 9 19 0 0 ==> Exponent value of non-zero matrix of 383-th row

Embodiment 3 1024 768 => Matrix Size (# of columns = 1024, # of rows = 768) Column weight distribution Ex) 0-th column weight = 14, 1-st column weight = 14, . . . , 255-th column weight = 13, 256- th column weight = 17, . . . , 275-th column weight = 17, 276-th column weight = 1, . . . , 1023-th column weight = 1. 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Row weight distribution Ex) 0-th row weight = 41, 1-st row weight = 40, . . . , 19-th row weight = 41, 20-th row weight = 5, . . . , 767-th row weight = 5. 41 40 36 41 41 41 41 41 41 41 41 41 41 41 41 41 38 41 37 41 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 Location of non-zero matrix (circulant permutation matrix) at each row 3 4 6 8 28 36 48 57 60 62 69 84 93 110 115 122 123 126 134 136 137 140 151 154 162 171 175 176 179 181 186 191 192 199 202 215 216 222 223 224 256 ==> Non-zero matrix location of 0-th row, row weight = 41 1 7 20 22 24 29 30 31 55 78 81 83 84 88 93 94 95 101 102 116 119 120 125 131 133 140 155 156 168 176 177 187 210 225 230 236 247 252 256 257 ==> Non-zero matrix location of 1-st row, row weight = 40 19 27 33 44 49 53 57 59 65 67 71 83 84 87 92 93 104 110 118 121 129 136 143 145 149 164 169 207 211 216 218 223 252 255 257 258 9 10 20 31 35 36 45 46 56 58 67 72 75 80 91 94 106 125 130 132 134 137 141 142 148 153 161 166 174 178 187 189 193 199 207 212 213 225 238 258 259 17 18 19 23 29 34 37 41 48 62 68 73 81 85 90 106 113 115 127 136 138 144 147 149 157 162 164 167 168 171 174 185 191 194 205 209 225 226 227 259 260 2 4 12 13 35 38 42 50 51 64 71 86 96 97 103 118 125 135 139 161 165 169 180 184 188 193 194 201 208 209 210 211 215 218 220 222 227 228 247 260 261 1 6 14 17 21 36 44 64 66 68 75 76 80 85 87 95 102 108 114 131 138 147 151 158 170 172 176 180 182 184 195 198 223 228 233 242 249 252 254 261 262 2 5 9 22 25 27 37 46 50 59 62 70 73 82 86 95 96 97 101 102 117 124 142 143 149 159 173 179 189 197 198 200 220 231 236 237 240 243 246 262 263 0 28 29 35 40 44 54 58 69 74 77 79 83 94 98 100 111 113 116 117 118 120 129 150 160 166 190 202 205 207 209 221 229 230 231 239 240 242 249 263 264 9 13 16 18 30 32 39 57 61 63 64 78 81 82 85 105 107 109 117 128 130 146 152 154 163 168 188 190 203 206 214 219 221 228 235 236 243 245 253 264 265 0 8 14 24 27 38 41 47 52 56 65 68 87 98 107 121 132 152 153 166 173 181 182 183 190 195 203 206 208 212 229 233 244 245 249 250 251 253 255 265 266 3 12 15 21 26 30 33 37 69 71 76 79 82 96 105 112 116 123 126 127 128 133 135 137 148 150 154 155 157 183 184 196 198 204 210 220 222 230 232 266 267 2 6 11 16 23 24 45 47 52 53 61 66 74 78 79 80 92 100 110 122 123 126 138 142 144 145 147 159 161 165 169 175 177 180 192 208 214 219 226 267 268 5 7 8 21 32 34 42 58 72 75 76 86 90 91 92 97 99 108 121 129 130 132 145 187 196 197 202 211 212 218 224 226 234 244 246 248 250 251 254 268 269 4 11 15 18 25 31 38 46 54 65 88 106 109 112 114 119 120 124 140 141 146 151 160 163 179 186 189 201 204 213 215 217 237 239 241 245 246 247 248 269 270 5 12 14 19 28 39 43 45 55 56 67 77 89 100 101 103 104 114 115 131 133 134 141 152 156 157 158 170 172 181 182 183 185 186 188 193 194 200 201 270 271 10 15 22 23 26 40 42 49 52 54 60 61 74 98 99 104 107 113 124 143 163 170 172 175 178 191 196 197 214 235 238 241 251 253 254 255 271 272 0 17 20 26 32 33 43 48 50 51 53 55 60 63 70 89 90 103 127 135 153 158 160 162 164 167 177 178 185 205 206 213 217 219 224 227 232 234 235 272 273 3 7 10 16 25 39 41 51 59 66 70 99 105 111 139 144 146 156 159 165 167 199 203 204 216 229 231 233 234 238 239 240 242 248 250 273 274 1 11 13 34 40 43 47 49 63 72 73 77 88 89 91 108 109 111 112 119 122 128 139 148 150 155 171 173 174 192 195 200 217 221 232 237 241 243 244 274 275 ==> Non-zero matrix location of 19-th row, row weight = 41 55 92 154 216 276 ==> Non-zero matrix location of 20-th row, row weight = 5 8 26 35 97 277 3 77 163 257 278 7 132 176 265 279 37 124 197 251 280 30 115 195 275 281 14 105 213 236 282 55 118 196 265 283 68 112 235 249 284 6 83 145 260 285 56 145 199 263 286 2 101 128 254 287 11 93 101 143 288 12 110 221 234 289 33 156 163 269 290 11 108 192 254 291 55 116 120 224 292 53 83 164 269 293 31 37 147 228 294 6 9 42 228 295 13 144 212 251 296 66 99 155 241 297 66 135 186 235 298 74 90 106 266 299 39 87 200 243 300 73 162 190 245 301 49 128 212 256 302 74 85 239 243 303 62 103 176 221 304 28 89 100 130 305 69 82 240 274 306 41 91 162 250 307 53 155 166 263 308 57 133 178 191 309 14 98 136 217 310 16 115 186 266 311 61 86 224 266 312 19 95 198 205 313 51 74 87 224 314 68 132 134 230 315 31 108 172 271 316 23 114 217 256 317 60 99 112 247 318 42 97 199 252 319 56 86 234 250 320 20 82 192 213 321 32 102 190 260 322 44 82 96 197 323 67 77 128 224 324 65 82 177 209 325 8 141 154 236 326 73 140 247 265 327 36 104 135 263 328 38 146 179 273 329 61 95 208 255 330 54 80 154 265 331 44 61 122 268 332 1 102 125 147 333 74 129 183 257 334 44 103 222 227 335 20 33 140 274 336 58 138 223 236 337 45 132 158 260 338 35 149 171 214 339 38 98 166 168 340 71 170 242 253 341 30 76 153 185 342 28 75 184 220 343 7 113 189 215 344 12 27 168 274 345 22 115 169 197 346 30 167 177 204 347 42 104 106 206 348 21 47 188 194 349 0 145 218 241 350 26 88 225 264 351 52 76 250 263 352 22 77 194 251 353 13 127 201 271 354 48 131 146 229 355 38 111 145 272 356 53 91 180 229 357 22 132 157 253 358 50 114 185 199 359 63 86 134 202 360 62 104 202 203 361 6 125 196 273 362 37 77 110 160 363 29 105 176 262 364 31 137 202 238 365 52 93 216 251 366 60 149 209 223 367 62 129 147 218 368 24 129 213 249 369 18 123 186 242 370 54 134 150 193 371 56 96 172 269 372 51 102 198 199 373 1 163 234 267 374 71 107 191 260 375 30 101 193 268 376 66 127 200 225 377 73 169 207 248 378 71 77 105 153 379 19 129 166 177 380 72 101 201 266 381 71 82 211 217 382 42 133 148 200 383 58 147 165 265 384 52 78 106 259 385 24 104 224 273 386 44 45 108 205 387 72 146 198 260 388 54 135 173 256 389 7 116 188 247 390 8 95 113 156 391 55 131 176 237 392 4 151 194 266 393 34 126 157 236 394 4 111 185 275 395 65 133 236 273 396 29 113 233 246 397 4 17 129 223 398 3 151 214 241 399 39 92 179 205 400 28 103 183 187 401 48 77 192 272 402 44 91 116 270 403 63 121 187 268 404 17 97 169 173 405 61 89 244 274 406 58 83 167 266 407 7 85 178 213 408 36 118 229 255 409 61 68 73 172 410 74 164 187 256 411 65 79 115 120 412 18 80 131 223 413 73 92 126 250 414 3 52 119 227 415 60 148 169 213 416 11 156 160 178 417 30 126 217 271 418 5 70 251 264 419 23 81 111 240 420 48 149 210 266 421 66 88 147 192 422 70 80 211 259 423 46 134 163 225 424 68 143 234 237 425 63 133 149 233 426 60 78 212 229 427 11 102 197 247 428 36 77 155 207 429 68 116 170 269 430 20 111 116 171 431 13 98 124 151 432 50 136 184 202 433 11 87 170 176 434 34 80 166 247 435 62 86 175 212 436 61 72 133 165 437 34 74 181 189 438 10 157 214 235 439 67 85 204 253 440 47 101 131 147 441 11 75 217 257 442 46 87 229 232 443 4 7 123 258 444 12 81 200 220 445 0 9 189 202 446 58 96 130 257 447 56 115 249 273 448 4 125 182 272 449 27 119 180 275 450 57 157 177 188 451 23 78 218 274 452 18 86 181 253 453 65 103 171 275 454 22 108 165 267 455 22 156 239 242 456 43 142 227 236 457 17 93 185 233 458 28 125 142 253 459 25 94 122 232 460 54 155 157 254 461 57 84 161 263 462 40 159 170 212 463 60 81 103 250 464 50 147 253 262 465 22 92 143 233 466 67 83 131 270 467 25 76 158 267 468 24 92 159 184 469 6 156 221 249 470 70 86 112 260 471 19 88 93 272 472 14 78 163 199 473 16 92 177 206 474 5 158 189 272 475 8 39 213 262 476 36 167 250 274 477 34 141 237 238 478 53 109 239 270 479 72 130 232 268 480 51 81 118 172 481 62 108 200 255 482 15 125 166 244 483 17 107 179 220 484 26 120 184 248 485 52 126 129 273 486 15 82 118 263 487 15 100 138 247 488 20 84 183 189 489 9 126 240 268 490 25 99 100 266 491 63 100 219 270 492 14 139 175 177 493 41 133 174 242 494 32 111 127 169 495 10 79 158 246 496 31 145 190 233 497 41 101 202 257 498 14 53 137 260 499 46 136 164 223 500 21 67 139 244 501 70 124 204 258 502 50 79 172 274 503 50 106 197 215 504 30 59 191 245 505 39 77 123 180 506 18 103 117 192 507 1 18 217 233 508 6 81 98 225 509 73 82 176 270 510 41 74 99 233 511 39 80 167 245 512 70 88 203 267 513 15 16 170 180 514 43 98 218 248 515 29 50 134 231 516 40 134 211 244 517 10 123 201 254 518 29 88 252 268 519 5 130 192 273 520 46 103 219 253 521 25 157 202 264 522 6 114 209 265 523 3 102 183 270 524 9 96 143 220 525 6 19 193 264 526 13 85 185 262 527 60 95 181 248 528 31 90 155 175 529 65 136 212 238 530 12 20 131 158 531 58 117 198 267 532 10 106 205 256 533 51 137 150 241 534 48 89 132 173 535 53 144 195 207 536 66 86 114 226 537 23 146 152 263 538 38 165 176 275 539 25 160 204 240 540 53 84 181 225 541 74 108 125 245 542 20 139 164 253 543 65 131 185 259 544 37 96 206 244 545 47 108 184 211 546 18 132 179 226 547 66 119 197 270 548 33 105 214 263 549 74 146 169 259 550 24 81 197 214 551 54 164 174 258 552 61 119 179 262 553 28 153 206 239 554 25 121 192 224 555 13 20 227 241 556 7 98 104 241 557 37 109 126 200 558 57 104 117 234 559 46 122 205 271 560 41 118 235 271 561 61 85 171 246 562 32 137 149 259 563 25 80 145 271 564 55 94 181 182 565 46 127 203 265 566 23 143 181 188 567 44 79 204 220 568 23 91 105 252 569 59 95 180 201 570 49 109 177 190 571 72 90 138 250 572 73 78 135 204 573 9 108 149 154 574 52 168 171 245 575 69 105 158 196 576 1 5 16 258 577 52 122 254 270 578 59 151 192 260 579 26 99 249 271 580 60 110 119 274 581 39 116 175 259 582 45 119 210 233 583 49 125 160 232 584 64 113 165 274 585 35 145 219 257 586 39 114 195 257 587 60 87 147 226 588 5 128 169 260 589 59 116 228 263 590 26 148 190 265 591 10 111 195 222 592 13 83 126 258 593 41 122 138 161 594 57 105 112 222 595 67 117 143 226 596 68 155 240 258 597 0 12 151 211 598 72 76 156 272 599 43 95 196 256 600 63 111 189 221 601 13 120 152 249 602 30 84 196 215 603 12 141 182 239 604 72 107 134 180 605 28 93 218 258 606 0 2 188 226 607 57 96 164 221 608 47 97 191 243 609 41 98 173 268 610 47 171 207 234 611 2 127 219 239 612 21 99 140 153 613 42 153 226 227 614 50 83 148 250 615 27 118 139 269 616 48 140 195 268 617 72 97 182 246 618 39 89 189 268 619 62 94 201 274 620 61 124 129 259 621 53 128 132 209 622 32 101 229 261 623 18 36 193 275 624 66 97 176 266 625 24 93 236 263 626 17 87 211 270 627 37 111 162 252 628 14 96 159 245 629 35 84 198 239 630 37 99 179 182 631 72 104 190 252 632 43 86 154 251 633 22 75 185 227 634 8 94 132 270 635 49 90 210 269 636 15 101 214 264 637 38 135 187 211 638 15 110 208 253 639 55 76 200 259 640 31 89 204 237 641 60 166 201 262 642 59 75 213 221 643 0 132 159 261 644 34 100 116 186 645 51 166 180 257 646 32 139 151 271 647 61 136 142 194 648 65 97 160 267 649 3 104 169 170 650 55 85 163 240 651 9 21 169 235 652 52 91 182 198 653 7 93 219 240 654 5 107 126 205 655 8 23 31 223 656 39 164 235 237 657 67 121 183 261 658 4 86 148 217 659 20 41 199 261 660 41 77 177 214 661 71 124 199 230 662 12 138 191 213 663 70 113 137 252 664 24 107 221 243 665 56 100 251 272 666 22 124 161 274 667 40 91 136 266 668 17 30 212 250 669 0 109 142 268 670 46 113 135 228 671 21 87 190 257 672 68 98 191 231 673 27 111 237 256 674 73 94 114 175 675 49 107 168 266 676 40 119 157 218 677 26 85 195 228 678 27 90 120 208 679 47 100 164 257 680 1 144 184 266 681 45 102 191 229 682 26 123 146 222 683 30 91 127 246 684 51 135 171 252 685 39 125 241 242 686 11 138 196 271 687 2 154 180 275 688 2 160 205 264 689 17 109 152 172 690 36 100 154 245 691 43 137 170 230 692 35 87 216 247 693 19 138 148 159 694 69 114 187 203 695 36 40 165 260 696 59 137 160 161 697 57 88 241 273 698 13 117 149 228 699 21 130 217 222 700 16 75 224 259 701 3 29 232 239 702 48 106 166 264 703 63 144 177 261 704 41 154 244 263 705 40 101 103 109 706 28 150 229 257 707 4 90 140 225 708 49 51 94 124 709 65 95 220 252 710 18 139 191 257 711 21 133 167 260 712 16 48 141 196 713 73 121 125 188 714 73 137 158 254 715 70 109 123 198 716 56 74 88 144 717 25 107 216 272 718 1 2 104 185 719 47 118 203 258 720 2 105 171 269 721 36 94 233 265 722 38 109 196 259 723 63 119 207 258 724 58 121 227 262 725 65 134 214 261 726 11 94 152 239 727 48 58 116 211 728 35 151 222 267 729 16 108 168 252 730 15 78 117 195 731 68 88 149 156 732 29 142 181 218 733 45 81 170 228 734 55 88 110 173 735 29 124 187 236 736 34 114 194 245 737 31 159 226 230 738 5 87 142 199 739 71 86 173 270 740 5 65 159 182 741 47 123 206 227 742 38 90 143 263 743 49 127 194 254 744 31 98 180 259 745 62 96 141 205 746 32 106 167 176 747 40 56 168 240 748 14 110 140 243 749 59 144 232 243 750 19 111 174 274 751 40 95 216 258 752 21 141 228 256 753 16 99 221 260 754 12 129 235 272 755 31 63 196 199 756 33 54 162 205 757 27 165 209 273 758 66 92 223 267 759 11 79 173 272 760 36 97 143 144 761 32 89 178 255 762 34 117 160 172 763 63 85 102 235 764 33 94 136 261 765 28 139 161 195 766 57 79 142 168 767 45 174 201 264 768 67 69 113 271 769 41 150 212 227 770 7 142 209 262 771 57 76 186 225 772 28 131 225 262 773 38 100 197 212 774 0 122 193 248 775 53 120 222 272 776 64 110 242 249 777 6 32 71 157 778 8 107 115 264 779 62 80 188 230 780 64 148 179 244 781 70 92 218 266 782 64 152 191 223 783 34 128 191 208 784 58 105 201 261 785 3 161 189 217 786 32 94 227 228 787 28 112 215 251 788 43 163 174 215 789 46 76 172 213 790 48 81 113 257 791 59 157 182 266 792 69 103 162 258 793 4 89 117 248 794 37 120 247 261 795 10 43 110 229 796 52 134 243 256 797 68 83 206 272 798 25 143 174 222 799 54 64 87 117 800 35 153 165 231 801 44 155 189 271 802 71 90 200 262 803 16 114 183 246 804 30 123 149 159 805 9 103 251 274 806 48 112 190 230 807 32 79 140 180 808 72 121 238 269 809 50 121 170 251 810 27 125 216 246 811 1 140 145 236 812 67 136 245 268 813 35 82 178 246 814 73 123 208 273 815 17 83 189 232 816 27 130 193 259 817 60 141 193 225 818 51 93 163 248 819 42 116 218 231 820 30 90 112 244 821 20 104 194 264 822 65 155 213 265 823 49 75 174 263 824 56 139 230 255 825 45 114 120 274 826 29 101 155 237 827 10 124 130 208 828 58 154 203 208 829 54 84 252 262 830 52 82 112 220 831 16 160 215 270 832 59 106 142 217 833 23 128 232 275 834 4 9 29 118 835 35 101 114 230 836 51 115 138 237 837 49 113 193 206 838 42 151 181 238 839 44 97 188 242 840 19 112 128 187 841 3 172 223 275 842 29 100 209 275 843 67 127 150 179 844 17 133 192 234 845 18 110 190 247 846 43 152 193 219 847 53 133 193 197 848 9 120 150 234 849 2 82 174 228 850 69 134 145 186 851 6 123 215 269 852 37 84 176 273 853 67 138 216 275 854 71 131 144 162 855 69 99 152 268 856 15 145 181 254 857 58 85 107 137 858 35 126 163 262 859 72 81 204 270 860 11 129 195 263 861 69 129 215 237 862 47 84 156 226 863 68 161 201 255 864 48 127 238 242 865 12 89 174 259 866 69 75 97 175 867 54 81 211 272 868 70 162 178 220 869 64 144 194 269 870 21 91 172 265 871 0 91 200 215 872 19 22 87 255 873 29 92 152 238 874 55 113 207 238 875 43 90 132 264 876 24 130 207 255 877 40 140 174 269 878 15 167 231 262 879 15 66 248 258 880 37 93 158 269 881 23 150 210 273 882 38 77 223 246 883 12 149 224 256 884 36 85 205 270 885 10 103 188 232 886 33 38 216 256 887 10 133 175 216 888 25 123 150 165 889 6 92 212 243 890 33 83 108 249 891 3 95 150 167 892 52 148 203 221 893 27 83 178 194 894 16 90 136 272 895 42 152 204 267 896 10 83 153 248 897 5 116 243 274 898 23 173 231 258 899 7 122 183 267 900 69 98 210 255 901 64 167 218 256 902 44 147 187 224 903 63 115 182 244 904 54 102 146 185 905 31 57 127 261 906 64 121 210 267 907 6 135 161 208 908 42 46 128 275 909 9 24 131 272 910 40 76 124 141 911 66 95 139 264 912 18 93 204 254 913 62 91 207 225 914 64 141 171 197 915 32 96 216 262 916 8 112 194 273 917 71 166 203 233 918 4 44 255 259 919 38 76 139 242 920 53 75 147 250 921 45 88 126 269 922 17 102 141 264 923 13 106 118 186 924 24 79 165 271 925 56 78 102 153 926 58 76 162 234 927 8 119 231 254 928 22 164 240 261 929 43 135 178 182 930 14 34 207 256 931 20 107 155 215 932 56 158 178 238 933 49 105 195 202 934 62 107 151 209 935 1 8 10 260 936 14 26 89 275 937 37 121 209 220 938 36 111 142 156 939 56 113 201 267 940 33 130 177 241 941 71 95 206 268 942 50 122 146 243 943 5 84 119 152 944 0 84 175 249 945 2 88 136 230 946 50 80 127 234 947 8 109 161 238 948 45 130 135 148 949 0 85 166 267 950 57 94 211 268 951 45 77 181 261 952 2 130 220 271 953 19 78 246 263 954 18 169 196 266 955 27 80 105 183 956 61 80 159 236 957 1 96 173 271 958 50 84 89 240 959 63 75 208 230 960 47 79 126 242 961 51 162 231 259 962 26 168 226 258 963 74 79 198 235 964 64 75 122 219 965 55 117 187 241 966 26 118 128 262 967 46 121 168 175 968 64 91 206 260 969 21 137 186 264 970 1 118 198 210 971 24 122 125 186 972 26 157 183 256 973 19 146 178 214 974 0 11 164 210 975 67 109 221 231 976 42 80 115 210 977 1 100 200 226 978 45 117 175 219 979 17 106 188 257 980 44 163 190 273 981 43 76 183 253 982 59 96 146 267 983 5 98 153 235 984 39 93 151 208 985 34 121 173 264 986 66 79 128 162 987 3 49 99 131 988 15 62 156 245 989 3 143 186 261 990 23 124 184 265 991 27 78 202 261 992 7 115 192 229 993 20 148 239 268 994 33 60 120 275 995 42 110 184 257 996 70 122 140 209 997 22 99 171 258 998 14 115 203 232 999 13 28 170 273 1000 4 158 167 184 1001 14 108 179 275 1002 12 92 153 265 1003 33 159 219 265 1004 51 110 185 222 1005 69 102 206 265 1006 35 104 179 269 1007 47 78 224 271 1008 46 89 144 267 1009 2 120 129 231 1010 70 106 219 249 1011 33 86 161 210 1012 34 82 150 261 1013 40 160 203 248 1014 59 81 187 247 1015 21 55 168 214 1016 13 25 207 244 1017 24 112 199 269 1018 68 75 198 270 1019 19 97 202 256 1020 7 78 154 222 1021 54 109 184 237 1022 ==> Non-zero matrix location of 766-th row, row weight = 5 9 119 138 260 1023 ==> Non-zero matrix location of 767-th row, row weight = 5 Exponent value of non-zero matrix at each row 31 16 2 9 10 3 18 28 18 3 13 14 24 25 11 2 6 18 24 27 3 8 16 1 4 9 30 6 5 19 19 9 25 0 24 27 31 0 7 0 0  ==> Exponent value of non-zero matrix of 0-th row 30 5 27 25 17 5 5 0 13 27 0 3 3 15 8 20 25 23 6 16 31 30 15 31 23 4 14 12 20 7 30 22 6 15 29 21 20 17 0 0 ==> Exponent value of non-zero matrix of 1-st row 11 22 29 29 17 4 21 7 0 13 16 1 13 20 18 19 1 16 27 5 12 15 27 20 25 27 11 17 28 31 0 11 23 24 0 0 10 12 22 23 8 19 30 31 4 29 24 12 21 9 12 12 11 17 10 6 18 30 26 22 9 6 0 3 17 17 2 18 16 19 26 18 29 11 2 0 0 2 31 15 15 20 15 24 16 3 23 27 22 23 27 21 13 0 4 6 16 5 26 19 12 20 19 12 17 10 7 7 16 4 25 28 8 21 21 8 0 0 14 12 8 28 6 1 14 10 13 23 18 25 7 13 22 20 9 26 24 6 30 11 5 20 28 30 27 23 7 13 20 14 8 20 13 23 20 17 3 0 0 0 31 26 6 29 6 30 14 19 22 20 30 17 20 2 1 10 27 17 0 14 25 17 22 14 26 28 21 1 24 6 9 3 14 29 30 28 13 21 0 0 16 17 18 5 25 10 12 23 1 26 12 16 20 2 20 15 20 2 9 3 30 19 8 23 23 8 24 0 25 14 26 19 21 13 2 19 13 14 19 0 0 30 5 26 10 0 28 5 1 21 21 16 30 6 6 11 30 14 17 3 13 28 22 7 24 25 13 15 1 11 17 1 22 24 30 22 4 15 31 10 0 0 7 4 1 1 5 26 12 24 29 6 19 3 26 17 10 25 12 30 23 18 25 13 12 16 15 31 20 28 30 27 12 2 4 16 20 18 8 28 21 0 0 10 11 13 6 17 25 6 26 21 11 4 28 25 0 1 14 16 13 25 14 20 1 11 10 5 31 0 21 18 14 21 27 9 18 12 17 22 27 4 0 0 16 1 28 20 4 25 30 17 28 16 22 15 20 7 21 9 2 12 31 3 13 0 31 13 19 30 24 11 29 25 30 19 20 5 7 22 22 25 28 0 0 3 9 19 20 9 0 7 3 27 28 13 14 18 4 13 3 12 23 16 7 17 17 2 15 27 13 7 19 18 27 1 0 0 18 22 14 14 7 27 0 0 17 10 31 19 19 12 1 18 2 26 15 7 5 8 22 25 21 3 4 25 10 16 26 25 20 16 3 17 11 8 23 13 9 30 26 3 22 6 13 0 0 6 4 15 2 12 28 1 20 7 24 21 1 10 19 23 28 10 26 27 3 1 21 29 22 25 17 31 2 20 31 13 15 21 17 1 25 5 19 4 0 0 18 5 0 22 11 27 1 17 13 26 8 9 17 24 19 22 10 7 19 21 17 6 25 11 16 10 17 11 28 29 18 26 25 6 22 18 10 10 7 0 0 4 8 20 25 1 2 6 15 27 4 0 22 7 0 9 3 0 23 20 12 4 18 25 1 0 16 6 0 13 7 2 11 25 2 26 24 0 0 12 17 20 11 19 2 11 1 30 3 22 15 7 13 31 10 20 3 3 8 30 0 30 7 18 10 8 12 24 30 31 15 19 31 7 21 8 13 16 0 0 30 8 26 13 0 21 25 16 29 2 24 2 15 27 16 19 15 7 26 25 7 8 5 1 5 27 5 20 11 0 13 23 20 27 0 0 0 24 31 26 27 5 16 23 25 31 30 0 9 15 13 6 24 3 5 19 10 12 2 15 10 2 13 4 9 9 11 14 16 20 21 28 22 7 8 27 0 0 ==> Exponent value of non-zero matrix of 19-th row 27 18 18 22 0 ==> Exponent value of non-zero matrix of 20-th row 19 21 14 16 0 21 25 7 0 0 29 5 3 0 0 23 0 8 22 0 10 0 25 0 0 16 24 11 30 0 15 17 21 0 0 27 24 26 23 0 5 25 23 0 0 29 25 13 0 0 9 31 14 0 0 26 11 8 20 0 23 15 15 3 0 30 29 0 0 0 5 23 11 2 0 21 8 31 27 0 15 3 9 0 0 4 29 3 17 0 27 19 23 18 0 23 23 30 7 0 1 5 20 6 0 1 17 12 10 0 22 14 30 0 0 21 10 15 2 0 21 6 12 30 0 6 2 7 0 0 29 25 31 16 0 1 19 24 6 0 26 25 22 30 0 23 11 4 0 0 9 28 15 3 0 17 21 15 0 0 31 4 10 5 0 1 21 19 22 0 5 31 14 0 0 29 15 11 0 0 14 10 26 20 0 23 10 17 9 0 8 8 2 19 0 23 20 15 0 0 23 3 4 0 0 7 30 6 24 0 21 19 9 2 0 23 13 29 5 0 17 14 7 30 0 3 11 28 0 0 1 25 13 14 0 5 1 22 5 0 2 2 8 17 0 25 11 3 30 0 21 10 14 0 0 9 17 30 0 0 24 19 26 0 0 15 31 20 23 0 28 17 18 0 0 6 21 17 0 0 17 8 11 28 0 13 0 5 0 0 12 6 24 20 0 28 27 4 0 0 8 14 27 11 0 4 10 23 0 0 4 16 7 28 0 29 14 15 0 0 2 15 30 26 0 27 7 11 23 0 22 30 11 11 0 8 8 17 5 0 14 24 26 0 0 6 25 16 9 0 18 9 28 14 0 8 25 0 13 0 15 27 14 28 0 17 5 29 12 0 13 13 22 0 0 12 7 17 0 0 17 10 3 26 0 0 28 18 0 0 2 10 30 12 0 30 27 28 0 0 31 18 25 19 0 6 28 26 19 0 30 15 20 7 0 21 12 22 24 0 3 28 21 15 0 30 17 9 0 0 26 8 14 26 0 17 12 9 0 0 21 25 9 16 0 3 8 14 10 0 27 10 7 0 0 22 4 23 27 0 4 23 7 28 0 30 7 29 25 0 28 29 16 5 0 8 9 27 0 0 20 18 12 16 0 27 5 21 0 0 15 15 25 0 0 12 23 0 0 0 10 10 28 3 0 10 17 26 21 0 19 27 9 28 0 31 12 17 16 0 28 16 0 0 0 27 26 30 26 0 28 6 2 16 0 18 28 28 0 0 20 23 24 0 0 3 11 11 0 0 14 13 4 19 0 28 19 9 0 0 1 28 17 0 0 20 22 30 12 0 4 6 21 23 0 15 27 16 23 0 26 28 12 0 0 6 6 14 9 0 21 13 27 29 0 26 21 15 0 0 19 1 23 6 0 17 26 18 0 0 0 22 6 14 0 29 11 31 24 0 30 10 28 20 0 19 18 16 0 0 23 12 3 0 0 9 10 28 0 0 21 15 0 28 0 10 15 8 0 0 19 1 25 0 0 30 21 12 6 0 7 9 4 30 0 29 15 28 25 0 7 1 20 0 0 4 12 26 15 0 7 9 21 7 0 4 2 23 31 0 2 22 15 24 0 12 22 29 8 0 20 22 17 9 0 27 5 21 0 0 14 21 26 0 0 14 27 8 16 0 30 21 22 0 0 19 14 24 28 0 27 11 8 0 0 26 24 14 9 0 19 15 0 19 0 23 4 29 0 0 0 24 25 8 0 25 22 0 16 0 6 8 2 30 0 31 15 15 0 0 16 31 19 0 0 13 29 24 23 0 27 24 17 7 0 19 24 25 22 0 19 19 0 1 0 18 13 2 1 0 25 4 31 14 0 16 11 6 1 0 23 20 19 30 0 22 20 27 18 0 22 2 6 25 0 18 31 28 0 0 3 8 29 2 0 23 21 25 0 0 7 25 0 11 0 27 8 21 0 0 0 11 21 0 0 5 3 14 0 0 11 6 12 0 0 7 2 24 0 0 19 30 22 20 0 22 12 9 0 0 5 9 3 23 0 5 13 14 0 0 11 2 4 0 0 26 19 7 4 0 22 18 24 10 0 23 21 29 17 0 31 8 3 3 0 31 9 31 14 0 26 28 21 27 0 12 1 24 0 0 27 10 4 19 0 3 12 20 7 0 26 9 22 0 0 22 17 1 26 0 31 18 1 0 0 14 11 27 0 0 3 17 8 13 0 20 11 19 0 0 26 28 27 0 0 26 21 18 0 0 4 0 4 30 0 10 11 12 9 0 6 31 21 0 0 3 21 24 0 0 22 29 23 0 0 6 18 24 22 0 27 2 29 0 0 8 1 21 0 0 14 22 17 21 0 29 30 2 12 0 27 22 31 7 0 19 18 29 26 0 25 7 11 4 0 29 17 10 0 0 20 5 18 0 0 31 12 26 28 0 6 6 1 17 0 9 20 12 0 0 9 18 11 0 0 18 20 29 0 0 3 29 0 18 0 24 2 29 18 0 21 23 31 15 0 18 0 4 1 0 27 10 23 22 0 20 27 20 0 0 10 25 9 0 0 2 23 10 11 0 12 14 7 13 0 25 7 24 0 0 8 31 22 0 0 17 17 10 30 0 20 17 20 10 0 23 21 1 26 0 7 6 7 27 0 13 27 17 26 0 16 14 6 26 0 27 13 5 0 0 4 1 18 20 0 30 30 21 26 0 31 0 0 0 0 10 2 6 3 0 1 25 7 9 0 17 4 23 10 0 31 2 5 3 0 9 8 8 3 0 12 16 27 0 0 8 5 19 0 0 6 9 23 25 0 0 21 13 0 0 16 18 28 0 0 0 28 29 0 0 11 27 14 31 0 30 29 24 0 0 18 18 30 0 0 0 21 25 22 0 19 15 17 26 0 28 17 23 9 0 21 27 24 6 0 29 2 8 0 0 6 10 16 0 0 22 13 1 13 0 0 20 4 29 0 3 14 11 10 0 22 10 22 4 0 23 27 2 0 0 5 6 23 0 0 26 25 17 29 0 16 30 15 27 0 12 9 17 15 0 4 10 22 12 0 13 16 30 0 0 30 17 16 1 0 18 1 2 1 0 16 13 12 29 0 10 17 3 0 0 8 26 18 0 0 15 25 7 0 0 10 15 4 8 0 0 29 9 0 0 12 28 19 0 0 11 25 25 14 0 4 20 28 17 0 23 0 4 13 0 6 10 2 0 0 5 5 11 28 0 15 17 11 10 0 23 12 28 0 0 14 2 30 0 0 24 13 19 6 0 12 12 6 0 0 24 13 26 0 0 1 21 15 17 0 8 31 20 0 0 25 17 5 2 0 14 23 2 24 0 2 18 3 1 0 6 8 23 10 0 6 28 12 17 0 18 15 8 31 0 4 7 21 24 0 31 11 17 15 0 20 18 4 31 0 11 27 3 24 0 0 30 20 0 0 23 12 1 0 0 19 25 6 0 0 16 20 11 0 0 9 7 11 0 0 24 24 18 0 0 9 4 31 23 0 7 29 27 12 0 8 11 29 23 0 1 25 20 0 0 11 1 29 14 0 5 8 8 7 0 14 22 21 0 0 6 19 4 0 0 22 8 1 0 0 28 19 24 30 0 25 20 10 0 0 3 22 23 26 0 14 11 19 15 0 19 12 4 5 0 20 3 22 24 0 11 17 29 21 0 10 1 8 0 0 4 8 1 0 0 26 18 20 12 0 22 10 2 12 0 11 11 1 15 0 30 23 12 30 0 9 15 22 11 0 4 25 9 23 0 9 3 22 8 0 0 4 25 19 0 0 26 25 11 0 22 29 16 0 0 23 29 4 20 0 7 23 29 23 0 23 29 12 9 0 22 19 19 30 0 7 30 17 2 0 7 26 8 0 0 30 17 8 0 0 3 3 30 5 0 23 2 20 0 0 1 9 28 0 0 18 14 16 0 0 26 25 23 29 0 30 16 19 0 0 25 26 14 0 0 26 13 27 0 0 17 23 8 0 0 25 9 31 0 0 28 26 16 1 0 10 19 24 30 0 0 10 2 27 0 0 8 26 20 0 2 18 6 20 0 29 27 6 2 0 2 23 12 30 0 9 19 5 0 0 13 16 17 0 0 0 17 0 0 0 30 19 28 19 0 6 26 6 20 0 9 31 29 0 0 5 13 6 29 0 7 6 23 0 0 1 21 0 1 0 24 18 21 0 0 11 27 4 11 0 31 30 1 0 0 9 11 4 0 0 8 23 7 24 0 20 5 24 0 0 26 14 25 21 0 23 31 12 8 0 29 16 21 23 0 12 11 5 4 0 16 2 21 17 0 15 19 29 8 0 7 11 7 26 0 25 8 4 6 0 18 3 23 0 0 28 10 14 12 0 3 30 26 0 0 11 24 9 29 0 13 24 21 24 0 12 31 13 0 0 3 18 17 0 0 29 26 16 25 0 14 18 14 0 0 17 1 3 0 0 29 27 17 0 0 15 13 28 27 0 10 5 15 31 0 20 14 18 26 0 23 17 31 0 0 16 2 16 6 0 16 2 17 0 0 6 14 4 5 0 22 22 12 0 0 18 24 15 3 0 18 18 29 13 0 22 25 9 1 0 16 14 4 0 0 30 15 14 0 0 9 1 12 5 0 3 28 5 2 0 24 24 12 3 0 28 6 11 5 0 20 22 2 22 0 22 23 4 16 0 4 24 29 30 0 29 17 9 0 0 21 1 26 1 0 10 24 20 20 0 8 30 19 9 0 30 23 7 31 0 13 8 10 14 0 1 6 4 28 0 18 27 25 7 0 20 18 19 0 0 0 10 27 0 0 2 31 10 26 0 30 13 8 27 0 11 23 1 10 0 23 1 17 12 0 24 15 28 4 0 24 10 17 0 0 23 28 11 12 0 15 29 4 5 0 30 12 11 0 0 4 12 27 8 0 27 9 1 19 0 4 30 14 30 0 27 6 9 3 0 30 6 2 29 0 24 14 7 0 0 25 19 13 21 0 10 0 5 27 0 24 9 1 19 0 23 22 25 24 0 18 9 17 5 0 14 17 19 29 0 28 17 11 0 0 11 12 1 0 0 7 18 9 7 0 12 29 14 11 0 4 19 1 2 0 19 12 23 0 0 27 14 29 6 0 30 8 29 4 0 0 3 19 6 0 13 7 6 0 0 15 9 20 16 0 29 1 2 27 0 31 19 8 24 0 10 8 5 27 0 24 27 4 1 0 27 22 9 8 0 3 25 6 15 0 22 22 13 9 0 14 29 19 22 0 10 7 16 21 0 25 23 2 27 0 2 13 30 27 0 4 8 25 16 0 13 28 13 0 0 31 25 5 14 0 29 5 1 0 0 18 18 19 9 0 26 24 3 28 0 31 11 11 17 0 10 13 25 1 0 14 13 16 0 0 2 0 7 13 0 10 16 5 0 0 20 6 25 0 0 7 13 18 0 0 22 15 11 13 0 11 15 10 12 0 5 2 13 26 0 12 19 5 14 0 21 29 19 0 0 22 11 22 22 0 24 11 23 29 0 8 25 4 15 0 14 16 31 17 0 24 25 26 28 0 4 28 21 0 0 4 24 7 30 0 1 17 20 23 0 27 25 15 21 0 17 2 18 10 0 28 3 22 0 0 1 28 30 28 0 5 12 21 22 0 9 21 20 0 0 19 15 19 6 0 8 27 9 19 0 30 22 18 17 0 5 15 12 27 0 25 30 20 31 0 19 1 15 7 0 23 23 25 11 0 12 27 6 28 0 21 9 16 5 0 7 16 17 0 0 24 24 23 28 0 16 17 5 0 0 18 23 20 0 0 28 1 29 25 0 30 16 17 0 0 27 15 5 10 0 19 1 5 13 0 11 3 28 25 0 31 4 2 20 0 14 27 28 0 0 6 5 17 15 0 4 4 4 25 0 27 15 2 30 0 16 15 22 0 0 18 26 12 0 0 31 22 0 21 0 29 27 28 30 0 1 24 31 12 0 29 16 2 3 0 31 4 21 15 0 23 14 21 19 0 19 28 1 9 0 23 10 18 0 0 4 26 28 30 0 29 1 5 8 0 4 30 20 0 0 6 28 21 26 0 4 14 25 22 0 21 28 21 23 0 1 0 16 17 0 2 20 24 3 0 31 18 8 21 0 7 20 14 11 0 19 10 19 6 0 10 16 12 13 0 8 19 26 1 0 13 15 12 28 0 15 23 11 29 0 12 22 20 9 0 1 27 1 0 0 29 29 16 30 0 15 25 10 24 0 9 10 7 0 0 3 5 15 27 0 10 21 13 9 0 8 10 25 5 0 14 14 26 0 0 24 6 19 29 0 10 5 7 0 0 2 13 29 10 0 2 30 1 0 0 3 4 17 8 0 21 31 5 23 0 17 4 4 17 0 11 21 10 15 0 9 5 12 15 0 17 22 25 26 0 20 25 18 24 0 18 0 15 0 0 23 18 3 8 0 11 18 8 13 0 26 27 19 31 0 30 29 20 21 0 26 27 13 23 0 8 20 28 9 0 12 27 13 28 0 6 6 14 26 0 28 18 7 30 0 14 20 6 0 0 27 7 6 17 0 8 10 6 4 0 28 13 12 22 0 8 8 23 0 0 30 11 29 29 0 13 21 8 31 0 5 14 28 0 0 10 0 7 0 0 11 1 28 0 0 18 1 14 4 0 15 27 17 2 0 4 23 6 30 0 16 22 9 24 0 7 20 27 0 0 16 3 14 9 0 9 5 28 0 0 1 28 29 26 0 3 3 3 0 0 24 15 4 0 0 20 16 7 10 0 9 18 0 13 0 4 13 18 9 0 31 21 10 17 0 12 20 13 19 0 6 0 20 21 0 9 20 5 0 0 26 23 19 0 0 24 17 14 0 0 5 5 12 14 0 25 14 3 7 0 27 17 2 10 0 27 27 25 31 0 30 8 10 0 0 26 18 21 0 0 12 8 21 0 0 20 22 14 23 0 14 9 21 6 0 29 1 26 7 0 6 5 4 17 0 3 19 23 29 0 5 30 7 28 0 13 19 19 5 0 25 1 12 0 0 24 27 21 0 0 28 0 31 3 0 5 0 25 0 0 27 27 7 23 0 7 29 30 0 0 2 25 19 8 0 15 23 13 0 0 4 13 16 0 0 15 9 27 20 0 27 6 26 9 0 15 29 4 13 0 11 17 10 4 0 29 13 18 16 0 15 25 25 0 0 29 19 22 0 0 22 26 20 8 0 17 17 11 0 0 5 1 26 15 0 30 22 9 4 0 16 3 22 20 0 6 1 12 0 0 3 2 25 0 0 30 21 16 9 0 3 28 6 0 0 18 4 16 16 0 19 24 23 0 0 30 16 27 14 0 10 10 4 0 0 28 10 31 20 0 23 11 28 0 0 20 17 25 27 0 20 23 27 0 0 23 7 26 17 0 5 0 30 0 0 19 18 0 15 0 30 17 9 0 0 28 1 5 28 0 30 5 30 21 0 24 11 12 15 0 31 12 20 28 0 3 6 0 0 0 7 4 15 19 0 27 6 17 7 0 21 13 27 1 0 5 31 19 0 0 20 16 7 10 0 24 29 22 0 0 16 21 6 21 0 3 5 6 30 0 5 5 20 12 0 5 14 22 16 0 28 17 15 9 0 3 17 4 7 0 23 18 20 31 0 5 11 26 5 0 2 6 3 27 0 5 5 4 30 0 10 4 5 17 0 0 28 0 0 0 25 18 30 0 0 18 13 15 18 0 1 8 15 21 0 30 27 16 0 0 29 9 5 27 0 8 23 16 19 0 28 18 16 1 0 23 23 20 0 0 30 29 11 0 0 22 24 2 23 0 22 10 13 21 0 27 7 22 1 0 9 22 9 0 0 7 2 3 29 0 2 27 15 0 0 24 10 3 0 0 6 21 14 3 0 29 2 7 2 0 6 10 11 0 0 7 27 4 24 0 18 19 28 23 0 20 31 27 4 0 26 0 20 2 0 25 1 18 15 0 3 23 7 16 0 23 5 25 0 0 30 22 24 0 0 8 24 29 16 0 9 27 1 0 0 26 6 26 16 0 13 6 31 31 0 29 7 9 0 0 8 19 30 9 0 25 7 4 29 0 20 29 18 24 0 22 21 22 0 0 5 18 24 18 0 6 14 23 0 0 20 20 26 19 0 2 3 3 0 0 10 14 25 0 0 30 28 11 0 0 16 16 28 28 0 22 23 7 0 0 14 15 15 23 0 1 22 17 0 0 3 11 28 5 0 23 29 4 0 0 19 26 27 4 0 10 17 1 0 0 5 1 11 19 0 18 1 29 0 0 31 7 20 0 0 12 26 0 0 0 0 15 11 0 0 18 17 30 25 0 20 20 29 5 0 8 15 6 25 0 24 2 3 0 0 17 0 25 12 0 12 25 18 7 0 14 7 29 23 0 5 6 20 1 0 22 11 15 0 0 16 27 18 0 0 26 8 19 0 0 14 20 16 10 0 2 20 29 26 0 ==> Exponent value of non-zero matrix of 766-th row 9 23 9 0 0 ==> Exponent value of non-zero matrix of 767-th row

[Embodiment 4] 2048 1536 => Matrix Size (# of columns = 2048, # of rows = 1536) Column weight distribution Ex) 0-th column weight = 14, 1-st column weight = 14, . . . , 511-th column weight = 13, 512- th column weight = 17, . . . , 551-th column weight = 17, 552-th column weight = 1, . . . , 2047-th column weight = 1. 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Row weight distribution Ex) 0-th row weight = 41, 1-st row weight = 41, . . . , 39-th row weight = 36, 40-th row weight = 5, . . . , 1535-th row weight = 5. 41 41 41 41 41 41 41 41 41 41 41 41 36 41 41 41 41 41 41 41 41 41 41 41 41 38 37 36 41 41 41 41 41 41 41 38 41 41 41 36 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 Location of non-zero matrix (circulant permutation matrix) at each row 4 5 6 20 29 32 62 69 76 82 87 93 99 141 156 178 199 210 218 235 239 265 284 310 318 335 410 417 432 445 446 456 460 467 474 477 499 500 503 504 512 ==> Non-zero matrix location of 0-th row, row weight = 41 11 15 21 33 34 39 42 47 77 105 110 125 130 153 206 209 214 249 252 268 281 294 298 320 329 355 377 382 404 412 426 430 433 440 446 448 459 463 483 512 513 ==> Non-zero matrix location of 1-st row, row weight = 41 1 9 28 56 60 66 80 91 99 112 120 151 171 176 188 190 201 204 207 209 229 269 282 295 296 298 349 371 372 408 422 431 435 439 461 495 496 498 506 513 514 11 19 41 55 65 84 86 93 111 117 129 139 140 146 155 164 192 208 221 234 247 276 294 299 318 370 378 389 392 418 419 421 451 467 471 473 484 486 491 514 515 15 28 37 40 44 73 102 104 122 124 137 156 160 179 180 189 207 221 235 246 258 270 302 304 307 346 352 356 357 365 366 378 380 393 409 431 457 458 469 515 516 3 54 74 77 82 101 123 127 128 142 148 154 157 175 184 192 198 216 223 229 253 278 286 291 292 294 306 348 359 361 398 402 424 436 453 456 463 492 501 516 517 9 18 22 33 38 53 58 76 79 84 94 96 103 137 162 177 195 205 257 263 268 271 292 308 311 312 321 347 352 354 365 366 375 388 423 429 437 441 453 517 518 16 42 49 70 80 82 92 126 136 139 159 168 174 179 182 183 208 217 228 288 298 328 343 344 363 369 376 390 395 396 428 435 438 444 447 451 458 471 473 518 519 2 11 52 60 61 104 109 127 134 138 150 161 162 170 211 229 240 253 275 283 292 296 304 305 331 341 345 367 382 387 398 407 411 421 462 474 481 498 504 519 520 0 5 12 18 37 39 85 111 114 121 131 136 148 187 233 238 259 269 280 287 313 336 342 374 397 401 405 431 447 461 473 478 479 481 482 490 496 501 502 520 521 7 21 35 57 94 99 119 123 129 134 152 163 176 183 193 200 207 213 226 261 278 279 293 303 305 324 328 330 359 361 384 400 413 424 445 465 472 475 478 521 522 16 27 34 50 53 65 69 71 74 81 113 121 124 145 170 171 193 194 243 244 260 277 279 310 339 353 374 379 381 390 403 421 450 464 481 483 497 505 508 522 523 1 12 20 46 47 72 79 97 109 115 116 122 157 158 192 200 202 225 232 261 360 364 373 377 381 390 392 415 441 466 485 493 498 502 523 524 32 35 54 68 90 120 140 150 172 173 186 194 196 205 206 217 221 241 243 283 296 309 349 354 358 386 393 405 410 414 416 422 442 443 475 477 484 492 494 524 525 6 8 28 31 48 71 88 118 153 168 177 202 218 225 227 231 250 266 275 277 306 307 308 313 319 324 333 341 343 348 357 368 379 389 394 423 468 469 488 525 526 7 17 22 26 27 39 56 132 139 142 158 186 189 196 203 208 215 224 248 251 255 258 273 304 316 325 346 364 386 404 405 434 441 443 457 465 474 476 477 526 527 5 13 41 44 45 66 70 109 125 128 163 173 181 197 219 234 236 238 302 307 315 321 334 335 338 353 371 396 406 409 415 417 462 472 480 493 497 500 509 527 528 17 37 47 48 54 63 69 80 86 103 106 108 149 152 161 166 188 195 214 220 227 231 233 260 262 334 338 342 369 411 422 427 428 430 438 442 448 451 455 528 529 2 26 31 58 90 112 117 125 143 144 156 163 165 175 185 222 228 259 265 287 300 326 343 354 362 374 376 377 410 412 420 449 454 459 463 464 466 483 484 529 530 0 15 23 55 75 95 101 103 119 158 159 171 172 178 203 211 218 226 230 234 247 290 297 303 322 338 339 342 345 353 360 376 386 397 401 408 413 415 427 530 531 13 30 64 76 112 113 116 123 126 149 178 199 206 210 215 220 240 250 254 259 270 277 325 327 399 416 420 486 487 489 490 493 494 497 499 501 505 506 507 531 532 24 59 105 115 119 131 146 151 170 180 202 205 212 219 223 227 237 242 244 251 274 288 301 310 314 315 318 350 351 355 370 375 381 385 406 412 419 437 449 532 533 20 31 43 45 51 56 100 108 113 133 154 155 159 160 167 184 187 190 191 253 266 311 316 358 365 372 391 403 408 425 446 454 461 462 482 485 488 505 510 533 534 3 4 23 38 63 67 90 104 116 121 147 153 155 174 197 212 213 248 269 271 274 291 299 300 312 320 326 347 382 396 403 419 426 442 471 486 490 495 506 534 535 8 19 22 32 59 74 78 130 136 138 143 160 193 196 219 228 230 232 237 254 262 267 272 283 285 295 331 337 341 372 388 399 407 423 425 437 438 450 475 535 536 10 12 25 29 50 75 84 91 92 102 105 108 142 144 173 191 213 250 255 319 322 331 340 367 379 384 398 433 435 460 466 496 499 503 508 511 536 537 52 70 81 86 87 96 97 100 107 132 143 204 230 243 270 278 280 281 289 291 308 323 330 334 380 418 447 448 459 469 489 503 509 510 511 537 538 1 2 4 10 13 49 55 68 83 85 94 124 135 154 245 252 254 275 282 284 289 323 363 364 368 369 373 383 409 452 456 476 495 509 538 539 14 19 25 34 46 48 58 60 64 87 89 95 98 128 176 187 212 246 251 260 273 274 287 289 295 302 312 328 329 385 387 411 416 439 443 460 476 489 508 539 540 17 18 24 35 40 67 88 97 135 164 168 182 185 190 197 198 211 216 220 239 244 255 257 263 268 285 323 332 333 350 351 352 362 392 394 395 406 444 467 540 541 16 41 43 46 52 57 73 75 118 120 131 147 165 166 177 181 209 224 265 272 284 303 327 332 337 340 346 349 356 395 397 418 426 440 452 468 470 479 487 541 542 9 30 49 88 98 102 110 111 115 132 133 141 145 150 198 222 231 238 267 293 297 300 314 337 339 360 361 368 385 393 394 424 428 439 440 444 449 450 452 542 543 3 14 45 72 85 92 93 106 107 118 130 162 166 189 191 194 201 236 247 252 276 297 299 301 305 320 322 325 327 333 351 355 356 367 375 384 400 430 445 543 544 7 10 36 38 66 78 89 100 122 145 148 164 165 195 203 217 223 235 240 248 249 264 266 276 301 309 313 317 362 389 413 453 458 470 472 485 500 502 510 544 545 6 23 27 33 51 98 114 126 135 138 152 157 169 180 185 201 239 241 256 267 285 314 315 316 317 340 358 373 378 387 391 425 429 432 464 465 480 504 507 545 546 0 24 25 26 30 36 62 71 77 96 106 134 140 147 167 181 183 232 242 246 261 262 281 282 290 332 344 345 348 370 414 433 436 470 487 491 546 547 8 50 51 57 72 83 95 137 146 149 169 174 179 186 199 204 214 216 222 241 242 245 258 263 272 286 293 317 330 347 357 359 363 366 399 401 402 404 436 547 548 42 61 62 67 73 81 89 129 133 144 151 161 169 172 184 215 225 233 236 237 245 256 264 271 280 290 311 321 324 336 344 400 417 420 429 432 454 455 457 548 549 29 44 53 64 78 83 107 110 114 117 127 167 188 200 224 226 257 264 279 286 288 309 326 335 383 388 407 414 427 434 455 478 479 482 488 491 492 494 507 549 550 14 21 36 40 43 59 61 63 65 68 79 91 101 141 175 182 210 249 256 273 306 319 329 336 350 371 380 383 391 402 434 468 480 511 550 551 ==> Non-zero matrix location of 39-th row, row weight = 36 97 214 493 504 552 ==> Non-zero matrix location of 40-th row, row weight = 5 93 260 456 470 553 110 157 304 462 554 76 224 246 383 555 127 254 276 543 556 50 99 168 409 557 106 155 200 476 558 52 252 262 518 559 73 79 370 486 560 82 331 480 484 561 109 204 329 495 562 78 166 318 506 563 135 187 209 537 564 86 225 493 520 565 69 304 466 470 566 101 149 379 418 567 12 201 442 550 568 25 200 444 525 569 72 321 370 487 570 147 298 400 520 571 119 264 272 513 572 48 62 347 457 573 31 52 233 546 574 56 303 468 531 575 32 49 234 479 576 129 206 404 476 577 92 293 320 524 578 134 235 366 538 579 62 179 434 486 580 95 321 350 506 581 29 292 418 470 582 112 164 349 549 583 72 173 436 478 584 4 318 388 546 585 4 273 395 436 586 70 241 369 551 587 27 250 258 543 588 129 199 354 453 589 51 178 419 519 590 86 258 475 517 591 9 43 249 383 592 117 151 338 508 593 76 182 377 525 594 18 224 471 551 595 93 205 327 551 596 35 270 365 472 597 102 226 294 544 598 86 185 322 397 599 89 231 445 515 600 87 152 288 346 601 17 51 379 412 602 12 252 332 385 603 45 239 396 450 604 70 234 441 542 605 117 243 461 485 606 88 236 378 422 607 131 291 513 540 608 104 280 327 551 609 36 166 381 461 610 127 317 449 512 611 146 221 348 531 612 146 248 257 502 613 114 244 322 469 614 2 3 230 515 615 140 285 299 530 616 60 92 385 515 617 57 297 402 501 618 8 178 284 537 619 74 240 371 393 620 141 229 497 534 621 84 312 430 531 622 75 204 242 467 623 66 251 279 378 624 116 191 222 480 625 18 212 386 550 626 7 162 390 464 627 113 260 500 518 628 40 253 476 535 629 38 256 281 457 630 111 239 391 539 631 118 170 405 431 632 15 162 462 471 633 90 226 271 330 634 142 218 291 464 635 125 150 342 421 636 140 230 246 340 637 137 198 308 358 638 72 230 428 510 639 99 277 372 448 640 17 327 344 433 641 28 284 401 533 642 63 148 332 432 643 133 227 429 540 644 67 259 368 528 645 83 301 473 514 646 149 261 376 488 647 95 205 486 522 648 61 220 328 460 649 50 278 405 512 650 126 264 426 526 651 110 235 403 522 652 36 298 305 508 653 92 172 416 454 654 10 11 454 532 655 126 186 373 491 656 115 171 368 509 657 93 319 363 448 658 114 291 417 495 659 84 329 333 489 660 0 252 441 501 661 73 223 504 524 662 41 153 223 455 663 110 269 456 533 664 86 193 306 357 665 78 259 296 409 666 18 204 434 490 667 102 163 301 360 668 83 203 296 366 669 80 280 507 525 670 85 301 329 373 671 136 172 377 477 672 9 235 236 432 673 115 149 217 391 674 28 181 461 479 675 118 184 238 500 676 14 249 321 464 677 2 186 312 434 678 65 204 319 350 679 93 234 297 430 680 81 257 472 516 681 45 227 232 514 682 19 271 276 502 683 80 306 312 481 684 0 198 209 421 685 139 248 374 511 686 52 101 314 372 687 35 274 496 510 688 108 224 250 460 689 74 270 501 511 690 141 154 197 422 691 133 319 339 432 692 66 348 367 529 693 44 235 314 419 694 35 201 440 530 695 23 284 358 503 696 6 193 410 439 697 22 211 344 473 698 38 246 397 496 699 103 156 187 524 700 17 274 331 408 701 8 164 464 504 702 141 155 375 478 703 112 219 362 453 704 67 162 359 389 705 20 197 320 511 706 26 33 271 303 707 82 314 411 532 708 104 196 264 348 709 87 254 364 490 710 83 206 297 546 711 96 208 417 528 712 117 197 274 465 713 44 331 375 509 714 19 37 291 459 715 104 185 237 330 716 39 203 332 537 717 51 85 349 356 718 124 349 370 477 719 137 206 324 455 720 14 163 417 421 721 30 251 455 551 722 90 188 384 413 723 97 275 438 527 724 119 167 331 460 725 67 93 283 381 726 51 188 267 366 727 118 223 246 277 728 23 123 166 263 729 132 162 500 516 730 124 292 367 511 731 26 157 382 486 732 109 255 298 479 733 27 217 462 515 734 141 234 424 433 735 117 234 268 544 736 53 105 285 342 737 30 245 388 449 738 133 157 291 482 739 79 178 367 541 740 101 299 466 509 741 48 171 303 547 742 112 225 288 472 743 116 217 427 526 744 57 217 315 530 745 115 233 316 544 746 4 195 242 285 747 49 206 377 504 748 95 150 373 478 749 12 269 375 525 750 83 158 320 487 751 94 225 300 528 752 114 262 337 519 753 126 214 318 437 754 19 270 401 506 755 61 168 216 529 756 23 331 492 541 757 85 192 475 536 758 98 163 442 518 759 13 339 352 374 760 86 229 347 527 761 102 182 507 529 762 138 215 231 453 763 123 315 479 536 764 48 200 216 500 765 112 154 251 544 766 92 170 346 541 767 9 270 299 523 768 16 233 433 490 769 84 153 404 445 770 85 176 263 516 771 89 227 355 361 772 90 169 260 551 773 23 31 265 289 774 137 168 408 441 775 125 220 316 524 776 49 192 409 492 777 8 347 515 544 778 85 200 353 465 779 103 139 282 346 780 62 196 356 380 781 137 228 378 489 782 40 212 365 492 783 36 184 201 476 784 25 220 386 472 785 8 180 506 541 786 4 202 410 493 787 37 265 458 489 788 22 225 367 403 789 28 147 371 471 790 43 212 368 407 791 19 161 308 461 792 140 180 313 512 793 146 207 384 437 794 124 216 330 463 795 131 237 357 544 796 108 276 481 488 797 99 187 438 548 798 56 129 267 393 799 58 171 280 542 800 32 229 332 545 801 55 231 309 533 802 54 160 336 348 803 48 261 344 437 804 106 178 350 424 805 20 177 269 491 806 43 289 425 430 807 86 197 310 530 808 91 296 418 507 809 79 241 335 396 810 87 242 492 543 811 125 263 383 390 812 13 158 378 392 813 57 256 306 487 814 134 283 439 465 815 139 353 388 452 816 38 150 189 501 817 44 289 388 518 818 45 223 430 517 819 26 212 485 493 820 46 59 239 461 821 107 337 371 391 822 36 53 287 355 823 70 329 437 482 824 77 235 328 369 825 31 237 437 538 826 120 274 364 522 827 47 189 458 484 828 139 163 317 520 829 87 174 296 466 830 80 173 233 483 831 28 209 304 329 832 56 203 387 433 833 71 169 480 533 834 116 336 411 539 835 143 184 247 527 836 146 152 382 519 837 13 308 344 435 838 46 183 306 520 839 32 202 338 368 840 123 253 322 518 841 145 305 307 487 842 119 150 269 546 843 0 152 312 422 844 69 158 476 521 845 59 231 351 439 846 64 197 417 516 847 9 174 248 526 848 6 200 325 454 849 106 268 360 436 850 14 168 472 526 851 89 256 327 391 852 74 181 195 498 853 2 263 271 534 854 118 207 365 456 855 135 328 364 501 856 82 186 467 545 857 109 152 420 494 858 131 235 324 546 859 47 307 391 547 860 25 229 501 543 861 135 158 269 358 862 37 269 351 413 863 126 229 451 537 864 20 24 355 356 865 146 327 343 506 866 128 259 421 452 867 117 359 401 526 868 31 186 399 519 869 44 192 398 519 870 65 226 227 341 871 128 265 389 448 872 113 153 278 377 873 73 175 470 539 874 149 184 417 484 875 30 189 284 545 876 4 196 305 399 877 99 290 394 495 878 96 206 362 460 879 6 236 311 537 880 115 240 426 482 881 31 224 281 292 882 140 257 380 470 883 5 11 250 379 884 53 204 248 419 885 37 113 173 304 886 82 112 290 488 887 12 172 334 475 888 111 243 463 499 889 147 292 333 439 890 10 212 216 520 891 63 156 241 548 892 60 266 366 444 893 38 208 238 450 894 135 250 393 550 895 42 221 354 491 896 143 187 279 420 897 75 241 446 506 898 53 154 327 501 899 130 233 307 375 900 15 294 362 549 901 97 153 276 509 902 147 187 415 505 903 66 157 352 508 904 78 245 302 517 905 4 7 371 533 906 48 293 503 508 907 124 199 308 386 908 102 310 443 519 909 46 201 470 515 910 28 235 341 536 911 135 192 385 431 912 140 185 422 471 913 138 273 413 517 914 16 213 250 512 915 86 167 438 545 916 147 202 252 347 917 111 196 449 528 918 92 202 272 278 919 120 208 455 543 920 144 164 247 525 921 49 246 488 490 922 106 225 256 314 923 34 250 312 408 924 92 260 394 420 925 92 251 269 296 926 103 255 356 543 927 64 345 371 513 928 103 191 445 449 929 104 183 510 532 930 47 172 467 527 931 106 165 456 535 932 13 174 412 510 933 76 189 315 468 934 139 194 285 532 935 80 150 356 541 936 81 187 399 483 937 60 213 446 541 938 101 177 282 290 939 137 151 414 526 940 149 300 330 530 941 57 233 383 443 942 143 250 474 516 943 116 159 414 430 944 138 276 316 416 945 22 219 229 360 946 49 210 418 423 947 101 179 251 500 948 147 215 254 318 949 87 200 316 370 950 13 190 287 516 951 29 228 345 405 952 13 316 396 436 953 46 220 419 535 954 59 202 442 464 955 143 171 425 456 956 108 237 431 505 957 79 236 418 535 958 58 257 469 492 959 75 198 399 488 960 12 21 188 327 961 131 211 466 535 962 121 175 451 469 963 97 216 374 469 964 16 55 277 357 965 96 182 322 474 966 23 97 140 486 967 142 297 469 534 968 81 258 295 340 969 58 215 454 537 970 30 32 398 538 971 18 218 340 443 972 56 217 237 338 973 71 212 255 524 974 89 152 406 474 975 58 67 388 393 976 38 302 456 485 977 58 334 479 499 978 99 179 266 414 979 3 261 461 536 980 43 66 86 512 981 126 188 331 483 982 97 195 218 405 983 62 207 308 359 984 41 242 290 474 985 120 222 385 485 986 30 253 344 549 987 41 194 437 485 988 11 163 457 543 989 93 156 357 532 990 25 300 336 362 991 45 271 370 443 992 91 209 444 523 993 42 173 477 529 994 24 324 480 501 995 18 144 421 544 996 50 152 235 491 997 30 309 331 434 998 10 42 57 333 999 91 128 316 468 1000 47 270 412 426 1001 39 195 302 543 1002 103 214 253 462 1003 134 217 244 259 1004 108 238 241 440 1005 104 313 429 542 1006 142 279 350 470 1007 84 198 384 521 1008 125 258 344 540 1009 87 255 350 534 1010 73 218 303 529 1011 12 263 422 512 1012 33 236 416 531 1013 13 28 170 242 1014 59 340 427 539 1015 65 207 222 541 1016 75 174 313 443 1017 65 67 422 520 1018 51 261 307 505 1019 78 233 442 508 1020 0 173 180 418 1021 93 210 503 525 1022 0 305 393 536 1023 100 263 463 542 1024 98 278 282 321 1025 140 267 376 441 1026 46 164 345 517 1027 62 209 484 526 1028 94 186 320 335 1029 79 203 492 513 1030 62 284 487 514 1031 22 252 359 531 1032 135 248 295 429 1033 107 231 374 541 1034 51 206 260 443 1035 64 182 317 341 1036 65 181 483 503 1037 87 336 455 523 1038 92 262 400 514 1039 98 183 362 451 1040 5 161 391 522 1041 29 282 385 467 1042 76 100 388 549 1043 81 213 431 499 1044 44 165 342 505 1045 28 155 379 481 1046 42 158 241 495 1047 76 167 369 537 1048 33 213 299 323 1049 102 107 449 484 1050 66 257 345 428 1051 122 222 247 288 1052 32 288 429 548 1053 73 162 372 459 1054 10 19 280 498 1055 128 195 365 466 1056 40 152 452 460 1057 52 259 344 432 1058 115 189 218 550 1059 24 267 373 507 1060 121 275 414 513 1061 111 184 296 338 1062 112 150 371 469 1063 88 206 465 550 1064 4 27 156 322 1065 81 155 315 486 1066 111 252 365 435 1067 31 40 283 350 1068 40 287 384 428 1069 130 211 380 408 1070 14 70 331 351 1071 88 344 484 538 1072 130 303 469 550 1073 24 275 315 404 1074 5 226 347 513 1075 45 158 182 309 1076 127 130 284 511 1077 60 164 432 521 1078 122 205 386 514 1079 85 277 458 486 1080 84 170 446 472 1081 95 257 259 396 1082 71 192 376 513 1083 35 191 407 490 1084 120 233 317 534 1085 16 162 230 326 1086 29 223 307 538 1087 13 50 387 400 1088 11 180 190 538 1089 114 205 278 365 1090 131 279 294 498 1091 53 239 468 551 1092 91 279 429 519 1093 18 225 354 484 1094 7 206 247 332 1095 24 171 389 477 1096 72 198 319 338 1097 52 189 439 492 1098 21 272 382 392 1099 141 161 302 536 1100 100 183 221 535 1101 44 214 390 501 1102 100 150 360 517 1103 30 176 273 536 1104 13 193 201 434 1105 90 151 501 512 1106 143 199 440 475 1107 123 249 418 420 1108 135 357 411 534 1109 91 178 262 317 1110 124 208 296 408 1111 118 231 288 516 1112 40 199 336 379 1113 16 338 496 530 1114 28 271 353 507 1115 67 243 416 427 1116 147 246 477 530 1117 148 177 245 534 1118 89 259 282 490 1119 7 286 298 380 1120 11 231 284 330 1121 1 219 400 481 1122 123 238 295 524 1123 20 172 312 323 1124 127 159 357 524 1125 78 182 355 441 1126 8 122 134 279 1127 136 149 333 361 1128 77 190 254 389 1129 74 221 395 537 1130 73 183 287 402 1131 103 176 451 502 1132 98 330 399 523 1133 117 186 369 540 1134 21 238 346 540 1135 77 160 476 537 1136 6 219 412 524 1137 106 156 298 538 1138 79 166 251 323 1139 50 304 313 347 1140 23 202 452 532 1141 28 191 458 540 1142 144 202 208 243 1143 49 194 306 360 1144 145 252 325 429 1145 113 268 407 431 1146 1 5 380 534 1147 29 55 297 442 1148 71 254 451 459 1149 6 24 457 463 1150 99 245 247 551 1151 29 181 369 516 1152 130 306 387 500 1153 17 165 193 504 1154 83 253 312 351 1155 43 83 303 319 1156 81 201 371 416 1157 54 192 327 347 1158 144 306 352 472 1159 47 194 273 521 1160 91 157 172 474 1161 105 228 515 543 1162 27 232 439 521 1163 0 188 254 360 1164 52 311 447 544 1165 50 167 283 335 1166 23 317 412 476 1167 27 220 356 488 1168 142 149 324 440 1169 102 223 511 548 1170 116 161 238 319 1171 148 265 384 414 1172 62 230 341 539 1173 52 161 223 530 1174 50 267 362 385 1175 64 237 490 518 1176 143 234 396 549 1177 25 358 448 473 1178 36 271 348 496 1179 129 154 380 427 1180 42 255 401 438 1181 97 287 408 475 1182 119 257 406 520 1183 132 234 336 457 1184 7 135 397 428 1185 75 217 294 532 1186 139 160 398 550 1187 40 226 264 442 1188 40 301 410 496 1189 44 185 296 495 1190 108 212 359 517 1191 58 174 272 333 1192 148 218 364 539 1193 40 332 419 512 1194 110 194 335 526 1195 122 219 265 322 1196 68 196 415 424 1197 138 213 377 386 1198 143 346 391 428 1199 144 250 343 378 1200 105 183 494 511 1201 144 230 396 505 1202 127 176 261 420 1203 89 189 310 551 1204 128 167 337 445 1205 16 91 404 437 1206 91 351 497 538 1207 97 321 389 516 1208 36 341 453 462 1209 124 192 404 512 1210 81 211 350 539 1211 105 211 265 297 1212 57 273 341 367 1213 133 294 389 494 1214 90 190 290 536 1215 133 186 371 473 1216 145 177 376 529 1217 67 337 376 482 1218 146 178 361 540 1219 72 159 314 406 1220 8 227 413 531 1221 75 160 413 540 1222 108 163 340 523 1223 31 170 328 493 1224 0 64 245 533 1225 25 194 251 271 1226 126 158 406 542 1227 104 254 473 475 1228 27 282 359 550 1229 55 191 328 355 1230 87 197 286 383 1231 129 228 294 438 1232 35 154 304 498 1233 60 261 518 548 1234 76 266 297 321 1235 131 255 258 531 1236 125 232 310 389 1237 94 261 392 464 1238 43 330 470 521 1239 34 189 228 488 1240 44 182 277 423 1241 2 218 433 451 1242 120 320 424 471 1243 94 329 334 415 1244 10 20 427 490 1245 104 247 430 522 1246 118 119 459 494 1247 113 201 280 402 1248 60 167 215 440 1249 122 237 369 533 1250 15 318 458 496 1251 15 220 260 298 1252 96 256 394 435 1253 89 183 377 513 1254 117 214 363 520 1255 79 252 329 406 1256 92 160 257 529 1257 9 39 352 404 1258 122 323 394 428 1259 63 281 372 421 1260 66 208 412 515 1261 136 164 408 487 1262 142 170 422 521 1263 123 222 311 460 1264 122 154 434 477 1265 0 244 437 509 1266 58 214 286 300 1267 10 159 366 453 1268 59 165 281 343 1269 128 223 465 547 1270 67 288 413 487 1271 55 169 391 535 1272 118 220 347 432 1273 56 175 420 525 1274 95 265 508 526 1275 135 181 416 451 1276 26 297 497 541 1277 107 171 213 493 1278 98 151 207 406 1279 138 210 311 459 1280 70 176 435 546 1281 68 211 316 526 1282 79 170 448 543 1283 84 325 397 499 1284 41 186 449 459 1285 39 222 291 551 1286 11 306 329 477 1287 9 175 245 488 1288 62 272 311 531 1289 56 209 307 520 1290 73 296 401 425 1291 82 339 395 431 1292 125 161 367 547 1293 113 245 382 520 1294 80 209 348 425 1295 90 249 323 548 1296 116 138 228 474 1297 136 215 383 509 1298 64 281 359 362 1299 136 200 378 493 1300 34 242 402 537 1301 60 219 399 507 1302 137 201 328 494 1303 3 302 366 540 1304 86 145 341 433 1305 106 290 489 524 1306 46 179 213 525 1307 136 207 371 517 1308 109 239 273 440 1309 75 256 459 514 1310 68 308 464 527 1311 21 166 336 510 1312 85 228 293 479 1313 28 249 426 463 1314 55 63 397 444 1315 34 182 325 546 1316 95 287 393 519 1317 130 151 334 398 1318 135 259 403 515 1319 58 223 299 407 1320 14 286 293 538 1321 39 60 71 221 1322 43 111 228 549 1323 140 253 425 480 1324 143 150 325 455 1325 121 155 248 547 1326 121 262 443 533 1327 62 206 346 410 1328 103 204 240 405 1329 20 162 278 529 1330 15 276 289 382 1331 147 157 328 484 1332 61 155 349 491 1333 72 172 259 343 1334 119 190 453 544 1335 53 243 424 521 1336 99 115 188 404 1337 119 191 275 412 1338 137 326 339 373 1339 27 38 127 272 1340 3 30 114 373 1341 33 118 317 482 1342 7 61 149 517 1343 78 207 397 524 1344 0 266 310 407 1345 70 165 466 481 1346 86 309 482 535 1347 127 166 425 534 1348 45 171 292 491 1349 100 262 349 396 1350 115 229 447 487 1351 107 172 411 470 1352 14 204 391 528 1353 138 165 252 255 1354 121 279 311 528 1355 107 282 426 505 1356 102 198 345 495 1357 70 245 450 530 1358 128 180 502 518 1359 64 205 249 438 1360 74 96 318 352 1361 21 247 405 517 1362 81 241 254 547 1363 74 210 368 502 1364 35 178 334 514 1365 108 277 323 521 1366 125 235 482 515 1367 52 318 434 524 1368 8 24 276 314 1369 80 322 449 532 1370 20 54 483 527 1371 132 154 315 413 1372 118 221 481 529 1373 36 290 351 477 1374 59 150 168 424 1375 114 242 429 496 1376 70 172 305 359 1377 138 245 387 551 1378 117 189 379 528 1379 10 227 338 516 1380 139 302 354 486 1381 148 152 390 535 1382 69 164 210 233 1383 19 204 292 487 1384 22 175 428 523 1385 39 173 337 439 1386 118 274 402 473 1387 105 286 467 525 1388 73 159 294 441 1389 89 214 411 442 1390 36 131 189 423 1391 69 284 431 544 1392 88 155 286 544 1393 68 248 381 400 1394 141 241 439 526 1395 108 121 351 467 1396 95 227 238 383 1397 48 290 398 544 1398 50 260 334 421 1399 0 251 497 522 1400 17 175 314 365 1401 2 328 409 426 1402 15 45 264 364 1403 139 201 244 275 1404 44 47 349 534 1405 131 263 339 406 1406 61 287 343 455 1407 99 225 402 452 1408 21 248 310 508 1409 4 227 374 497 1410 65 180 409 473 1411 47 181 253 447 1412 6 168 388 440 1413 93 179 353 450 1414 144 289 475 546 1415 22 166 337 548 1416 148 161 255 394 1417 54 302 495 531 1418 123 160 308 364 1419 54 213 349 418 1420 12 34 237 542 1421 82 185 393 400 1422 6 68 398 487 1423 98 232 326 343 1424 82 156 252 340 1425 97 279 450 534 1426 63 274 370 374 1427 132 180 282 386 1428 107 155 266 551 1429 95 200 272 515 1430 148 185 504 526 1431 3 25 405 484 1432 140 189 244 463 1433 42 331 410 468 1434 64 195 293 506 1435 109 342 387 547 1436 68 240 333 423 1437 116 267 310 520 1438 137 281 303 533 1439 114 168 416 523 1440 120 149 318 457 1441 95 162 309 348 1442 19 213 448 497 1443 110 228 370 520 1444 47 254 353 478 1445 5 214 444 488 1446 2 167 435 500 1447 22 116 353 516 1448 74 177 409 476 1449 34 141 403 532 1450 74 163 386 420 1451 141 208 253 406 1452 55 72 251 518 1453 126 202 430 490 1454 68 157 394 451 1455 94 166 416 536 1456 57 185 506 539 1457 64 103 326 370 1458 29 273 358 529 1459 24 218 272 472 1460 1 208 237 469 1461 18 61 382 532 1462 43 188 333 496 1463 122 188 358 500 1464 146 340 356 459 1465 77 301 377 469 1466 21 41 418 543 1467 50 196 301 540 1468 11 283 507 517 1469 53 233 445 542 1470 92 151 429 460 1471 136 191 348 499 1472 34 169 419 498 1473 74 239 504 544 1474 146 320 377 439 1475 54 273 337 545 1476 145 264 312 388 1477 41 309 447 520 1478 47 156 212 248 1479 90 246 373 445 1480 2 288 415 478 1481 61 74 334 521 1482 29 185 382 524 1483 37 202 423 543 1484 7 173 340 514 1485 112 283 414 515 1486 5 263 325 519 1487 15 186 236 365 1488 126 200 389 415 1489 2 160 266 528 1490 54 247 373 539 1491 66 324 387 537 1492 44 84 132 249 1493 138 151 280 543 1494 100 225 479 522 1495 41 168 458 531 1496 106 292 323 326 1497 63 184 299 406 1498 134 326 364 401 1499 21 124 328 497 1500 65 200 411 546 1501 3 108 297 482 1502 78 156 468 522 1503 142 163 461 550 1504 78 312 367 465 1505 5 193 413 426 1506 5 242 453 518 1507 71 243 262 509 1508 136 241 282 366 1509 101 193 390 547 1510 82 203 450 462 1511 83 343 386 534 1512 85 153 425 523 1513 27 163 361 538 1514 101 204 396 515 1515 88 176 270 424 1516 10 265 339 471 1517 63 353 377 536 1518 77 250 464 545 1519 134 184 380 483 1520 57 236 372 517 1521 114 238 294 526 1522 21 215 397 546 1523 12 33 389 461 1524 148 205 368 551 1525 33 225 253 374 1526 138 184 471 492 1527 71 95 206 292 1528 124 151 433 481 1529 53 175 300 500 1530 101 356 428 545 1531 26 197 447 454 1532 126 284 422 450 1533 79 267 395 490 1534 127 199 347 525 1535 20 214 350 435 1536 134 226 499 510 1537 67 165 322 513 1538 124 193 373 528 1539 110 227 408 444 1540 145 275 360 547 1541 37 169 363 523 1542 37 155 283 474 1543 102 253 333 521 1544 35 250 366 533 1545 109 216 372 514 1546 92 274 373 438 1547 13 213 367 380 1548 100 153 379 441 1549 24 156 384 542 1550 138 261 421 496 1551 55 56 126 403 1552 123 171 441 495 1553 137 175 255 289 1554 29 300 325 459 1555 69 104 198 420 1556 7 75 299 502 1557 52 157 446 540 1558 145 232 380 548 1559 39 308 409 529 1560 4 224 432 475 1561 94 155 384 417 1562 31 290 353 407 1563 102 178 202 481 1564 69 117 332 524 1565 45 176 363 467 1566 91 199 263 483 1567 26 338 372 426 1568 94 264 273 466 1569 121 214 316 409 1570 79 214 417 531 1571 26 230 403 528 1572 9 240 392 433 1573 72 211 270 348 1574 23 193 217 543 1575 33 164 463 536 1576 76 251 459 530 1577 35 295 326 467 1578 14 22 502 505 1579 113 227 401 542 1580 99 195 379 392 1581 10 244 467 479 1582 103 305 427 478 1583 77 211 280 346 1584 122 194 286 546 1585 108 193 354 508 1586 84 87 368 392 1587 105 156 382 547 1588 51 239 403 525 1589 59 222 242 549 1590 113 197 396 398 1591 134 179 291 374 1592 39 210 394 528 1593 105 349 368 540 1594 43 280 305 524 1595 144 267 351 518 1596 112 278 369 510 1597 14 258 313 449 1598 59 354 432 525 1599 57 157 231 538 1600 69 169 322 527 1601 2 210 217 342 1602 18 151 361 483 1603 69 275 277 343 1604 119 179 313 535 1605 112 239 344 441 1606 1 30 414 525 1607 111 181 281 384 1608 116 186 279 542 1609 35 277 406 485 1610 132 344 395 494 1611 96 160 390 542 1612 78 291 346 435 1613 82 205 231 380 1614 6 36 102 364 1615 132 248 301 408 1616 70 248 378 523 1617 72 193 489 549 1618 39 244 413 493 1619 31 243 502 549 1620 96 191 285 519 1621 15 198 454 542 1622 32 179 403 517 1623 145 159 394 472 1624 32 211 378 521 1625 39 176 184 519 1626 120 187 419 436 1627 45 225 431 550 1628 41 239 339 503 1629 3 187 264 377 1630 54 207 450 454 1631 58 209 291 343 1632 135 161 509 532 1633 124 288 345 395 1634 66 183 365 483 1635 34 360 432 530 1636 109 190 423 550 1637 85 231 321 550 1638 26 174 438 517 1639 38 355 430 547 1640 11 12 324 368 1641 110 258 325 436 1642 38 209 323 516 1643 85 217 269 548 1644 51 69 236 497 1645 16 163 385 478 1646 88 242 245 423 1647 59 265 438 522 1648 145 153 265 532 1649 8 277 278 539 1650 42 325 505 536 1651 109 172 229 303 1652 121 304 355 530 1653 2 178 313 447 1654 34 203 262 511 1655 43 162 444 532 1656 134 158 327 488 1657 37 281 348 404 1658 104 220 372 539 1659 42 90 388 462 1660 19 217 422 522 1661 140 158 361 545 1662 56 194 361 548 1663 17 228 337 507 1664 63 243 387 516 1665 4 211 249 450 1666 21 177 365 415 1667 122 151 353 509 1668 130 191 517 523 1669 49 171 435 486 1670 46 262 321 522 1671 137 262 363 397 1672 41 174 342 475 1673 64 159 301 521 1674 27 230 240 507 1675 66 268 335 533 1676 115 159 381 499 1677 32 161 476 514 1678 87 179 400 485 1679 9 11 410 534 1680 34 350 398 426 1681 141 289 295 405 1682 24 258 260 510 1683 29 48 302 342 1684 49 339 402 523 1685 139 261 362 423 1686 52 200 480 529 1687 9 23 417 547 1688 132 335 363 530 1689 6 195 376 527 1690 85 167 332 393 1691 59 197 393 547 1692 116 196 454 518 1693 98 203 356 527 1694 26 329 436 512 1695 30 221 321 411 1696 88 160 427 431 1697 5 194 281 541 1698 51 211 246 536 1699 88 180 215 303 1700 24 221 363 528 1701 83 177 243 345 1702 115 165 516 531 1703 63 168 284 456 1704 125 323 429 442 1705 126 308 382 499 1706 142 270 345 357 1707 94 221 322 328 1708 76 161 362 489 1709 123 179 232 371 1710 127 210 390 446 1711 20 238 400 468 1712 87 163 369 376 1713 98 215 410 516 1714 79 150 308 357 1715 129 306 444 533 1716 77 120 472 518 1717 7 226 489 535 1718 107 255 320 529 1719 14 158 469 498 1720 21 198 316 479 1721 17 226 478 537 1722 34 210 216 305 1723 96 153 378 399 1724 120 246 367 474 1725 23 183 266 539 1726 26 182 254 339 1727 123 300 432 458 1728 12 196 489 533 1729 90 256 293 395 1730 19 82 301 355 1731 100 233 295 444 1732 48 235 386 450 1733 56 321 466 547 1734 99 361 390 395 1735 113 181 288 535 1736 124 162 307 480 1737 98 240 448 491 1738 83 191 354 436 1739 25 177 452 538 1740 120 174 384 536 1741 136 315 357 550 1742 20 228 473 509 1743 47 98 443 538 1744 78 190 390 484 1745 114 199 465 539 1746 110 213 376 503 1747 42 178 324 416 1748 75 232 324 436 1749 117 173 287 546 1750 118 159 375 538 1751 95 275 334 399 1752 129 182 229 384 1753 15 81 296 414 1754 90 311 343 415 1755 73 187 413 546 1756 49 283 462 463 1757 55 280 352 465 1758 51 238 451 512 1759 17 157 514 520 1760 111 221 316 475 1761 68 232 289 532 1762 81 222 256 261 1763 32 256 275 482 1764 80 157 368 448 1765 3 303 385 412 1766 143 177 326 383 1767 106 244 503 507 1768 41 330 428 498 1769 134 224 385 402 1770 27 188 354 400 1771 37 199 295 541 1772 75 277 410 523 1773 97 304 468 499 1774 107 208 276 494 1775 142 192 264 383 1776 142 256 358 458 1777 15 181 317 439 1778 75 205 442 526 1779 61 110 234 389 1780 114 190 300 430 1781 35 247 307 381 1782 94 125 350 545 1783 63 180 381 489 1784 119 271 504 542 1785 1 310 394 521 1786 5 207 407 540 1787 5 15 433 440 1788 22 46 334 458 1789 83 219 390 544 1790 72 295 457 527 1791 72 94 203 456 1792 121 205 230 449 1793 68 169 170 477 1794 13 282 397 480 1795 101 312 461 519 1796 47 167 352 499 1797 1 156 311 337 1798 67 194 295 417 1799 110 180 399 496 1800 30 198 419 427 1801 48 183 427 523 1802 1 215 314 401 1803 129 170 276 530 1804 132 202 376 535 1805 100 172 341 526 1806 83 244 382 505 1807 104 292 416 445 1808 51 58 341 485 1809 56 77 326 434 1810 80 207 412 549 1811 139 150 446 539 1812 141 272 409 445 1813 31 209 239 514 1814 61 249 465 543 1815 0 6 278 542 1816 66 210 355 452 1817 149 255 306 442 1818 19 183 253 440 1819 14 215 285 455 1820 43 170 495 546 1821 105 158 381 460 1822 3 22 324 492 1823 54 168 317 547 1824 80 237 299 545 1825 16 293 409 549 1826 36 174 314 422 1827 129 313 452 524 1828 69 212 240 441 1829 96 175 268 447 1830 4 175 315 381 1831 132 207 298 528 1832 40 191 218 298 1833 111 246 386 548 1834 62 165 270 294 1835 53 249 360 474 1836 100 180 428 438 1837 9 309 402 540 1838 119 198 349 518 1839 102 234 372 468 1840 115 335 470 513 1841 23 250 320 445 1842 103 223 387 497 1843 70 154 220 385 1844 68 300 353 395 1845 17 25 292 531 1846 29 166 275 419 1847 131 162 229 285 1848 82 226 315 421 1849 12 302 460 529 1850 17 290 338 534 1851 46 301 383 512 1852 134 208 354 372 1853 18 182 238 463 1854 10 179 281 527 1855 129 273 319 375 1856 121 167 464 503 1857 140 220 415 513 1858 61 203 313 437 1859 105 136 260 492 1860 88 212 272 358 1861 46 226 394 437 1862 25 208 280 381 1863 130 234 464 538 1864 22 216 364 379 1865 103 358 481 510 1866 27 236 345 542 1867 80 218 366 494 1868 74 315 359 541 1869 143 266 318 514 1870 33 204 260 444 1871 58 154 420 508 1872 132 210 485 513 1873 81 152 447 477 1874 112 223 254 293 1875 32 73 305 397 1876 127 300 378 426 1877 50 166 338 513 1878 146 175 263 549 1879 111 154 242 505 1880 97 169 369 521 1881 16 176 334 374 1882 54 240 452 514 1883 107 195 329 525 1884 40 165 251 339 1885 106 274 414 446 1886 128 256 427 529 1887 130 174 327 335 1888 91 245 375 457 1889 33 192 346 533 1890 145 178 291 525 1891 128 247 360 493 1892 3 44 374 407 1893 53 185 370 545 1894 1 230 402 527 1895 90 176 355 454 1896 144 152 297 375 1897 33 224 356 522 1898 125 212 404 522 1899 114 164 364 528 1900 54 76 351 403 1901 65 203 310 446 1902 131 164 363 486 1903 128 151 235 537 1904 77 195 400 425 1905 42 362 466 533 1906 76 201 491 502 1907 55 258 424 425 1908 120 167 227 298 1909 57 171 405 545 1910 84 283 443 545 1911 8 161 436 527 1912 99 154 443 515 1913 1 170 259 548 1914 133 192 241 379 1915 60 226 447 527 1916 89 199 498 500 1917 19 188 234 337 1918 60 177 403 417 1919 93 109 222 370 1920 71 201 415 541 1921 31 205 479 527 1922 17 181 286 519 1923 52 206 498 548 1924 67 224 330 476 1925 2 224 420 480 1926 16 196 268 342 1927 149 219 257 435 1928 148 307 333 503 1929 148 268 410 425 1930 147 258 398 549 1931 37 220 366 519 1932 117 187 289 535 1933 144 181 453 495 1934 7 225 454 494 1935 115 319 401 519 1936 18 215 279 511 1937 121 184 448 545 1938 105 199 236 518 1939 122 236 446 550 1940 133 262 286 410 1941 128 241 330 434 1942 104 155 395 453 1943 94 285 408 448 1944 111 160 323 393 1945 113 142 367 467 1946 48 173 453 549 1947 65 159 340 423 1948 42 194 196 550 1949 101 310 317 537 1950 16 219 419 494 1951 56 164 336 407 1952 113 332 411 471 1953 89 160 392 473 1954 93 263 309 511 1955 71 271 278 474 1956 130 221 452 461 1957 45 267 342 346 1958 1 239 480 489 1959 6 246 319 513 1960 125 195 243 532 1961 105 237 335 478 1962 148 295 369 462 1963 84 222 294 491 1964 119 243 260 551 1965 49 152 232 412 1966 123 268 481 546 1967 69 298 302 548 1968 11 307 435 549 1969 9 106 336 462 1970 109 186 398 522 1971 62 153 440 478 1972 38 286 471 497 1973 147 153 203 358 1974 14 341 399 430 1975 3 169 197 457 1976 146 169 266 387 1977 108 171 285 288 1978 61 320 407 451 1979 144 176 468 513 1980 8 257 305 455 1981 68 285 375 510 1982 86 209 293 491 1983 18 196 418 545 1984 38 204 376 465 1985 129 268 482 506 1986 1 35 48 133 1987 39 257 375 458 1988 53 276 456 537 1989 63 232 392 449 1990 101 169 229 503 1991 73 232 331 347 1992 50 270 340 455 1993 127 244 460 531 1994 96 259 363 512 1995 76 216 258 551 1996 64 352 415 523 1997 98 199 431 531 1998 91 222 264 515 1999 33 173 414 541 2000 100 231 319 504 2001 130 185 404 502 2002 110 166 197 504 2003 77 165 313 363 2004 133 249 324 520 2005 107 326 423 540 2006 93 173 392 539 2007 41 266 269 359 2008 145 190 224 335 2009 37 174 247 293 2010 65 137 463 544 2011 136 230 424 429 2012 71 159 268 361 2013 71 78 392 471 2014 84 216 391 541 2015 59 314 352 405 2016 55 224 446 548 2017 45 80 332 457 2018 26 89 219 269 2019 131 153 274 536 2020 66 168 261 411 2021 139 185 283 311 2022 36 192 387 493 2023 88 287 447 466 2024 8 240 274 342 2025 49 299 318 351 2026 141 190 309 401 2027 109 184 445 506 2028 7 25 424 434 2029 38 146 388 473 2030 147 205 267 498 2031 116 333 352 456 2032 65 143 219 304 2033 32 320 421 528 2034 60 216 381 396 2035 20 28 289 357 2036 57 190 299 411 2037 133 244 433 535 2038 77 240 311 508 2039 70 218 349 514 2040 112 193 483 485 2041 133 177 287 512 2042 46 88 325 361 2043 96 188 304 534 2044 142 252 309 354 2045 10 336 512 522 2046 ==> Non-zero matrix location of 1534-th row, row weight = 5 11 187 345 530 2047 ==> Non-zero matrix location of 1535-th row, row weight = 5 Exponent value of non-zero matrix at each row 23 30 18 12 18 17 12 8 26 0 14 23 16 22 7 24 7 6 14 27 28 19 14 10 1 13 1 9 26 30 0 28 11 4 26 11 17 31 28 2 0 ==> Exponent value of non-zero matrix of 0-th row 17 13 14 28 15 20 30 1 0 0 26 13 1 13 0 19 9 8 27 2 28 17 23 6 2 3 20 18 23 23 25 29 4 6 12 11 15 14 31 0 0 ==> Exponent value of non-zero matrix of 1-st row 14 14 23 12 12 1 22 5 2 1 18 5 9 19 5 11 24 2 5 9 10 12 3 23 2 28 19 13 12 24 14 17 6 27 20 14 31 5 22 0 0 26 30 17 19 8 6 8 16 5 24 21 16 0 30 24 9 5 2 31 28 28 7 18 26 29 17 13 31 5 14 20 21 10 15 1 21 12 2 2 0 0 15 13 16 1 12 2 11 4 18 10 21 22 22 31 5 5 30 10 24 14 0 28 2 21 26 26 30 26 21 19 4 8 22 10 8 16 3 23 3 0 0 22 6 15 23 9 31 30 13 25 25 1 2 4 31 25 1 18 10 29 18 2 24 21 20 14 29 5 6 9 27 6 13 23 13 27 17 1 13 2 0 0 8 22 28 9 5 25 3 4 22 18 26 28 19 3 9 10 26 29 13 27 20 25 30 24 14 15 15 23 28 4 21 19 28 5 11 11 31 22 21 0 0 6 6 29 4 31 10 3 26 0 31 27 14 18 25 24 18 20 16 7 19 10 0 28 1 24 20 12 15 15 3 29 29 6 27 27 12 13 30 12 0 0 18 31 10 6 8 22 30 31 0 3 26 5 2 29 5 16 24 26 21 20 16 9 11 30 3 14 28 19 20 15 23 4 27 14 4 6 2 23 10 0 0 16 15 12 20 19 4 24 20 7 4 0 9 6 22 27 29 25 14 3 18 15 11 12 30 21 27 21 20 3 6 29 5 19 9 24 7 23 8 2 0 0 21 11 11 19 23 9 9 19 3 29 8 15 4 23 0 26 26 6 27 29 2 31 26 17 24 25 20 15 26 27 0 3 14 3 22 26 7 4 17 0 0 13 4 13 23 26 26 17 28 26 19 31 3 14 22 20 7 27 9 15 25 24 21 5 11 5 9 23 13 6 1 1 25 16 8 20 4 12 28 6 0 0 7 26 18 8 27 15 25 24 21 2 2 24 0 27 0 31 8 13 13 15 23 7 9 7 30 5 24 25 23 12 23 30 8 21 0 0 5 9 11 9 18 25 11 15 11 19 17 1 4 16 0 6 3 19 30 28 22 30 20 19 27 4 24 2 6 29 9 20 18 10 26 3 10 0 1 0 0 5 11 6 25 5 29 11 28 23 30 26 27 19 9 13 17 30 30 12 18 28 9 27 30 10 28 8 12 6 7 19 10 23 10 0 15 26 15 4 0 0 1 26 18 2 30 6 17 31 5 8 20 13 2 19 29 9 9 21 20 21 23 2 15 31 30 4 1 21 4 11 11 20 30 5 16 25 0 13 20 0 0 14 19 7 23 21 29 5 11 9 25 1 19 14 0 18 30 25 27 6 14 21 27 9 0 3 4 2 31 19 6 11 11 3 13 24 2 26 4 13 0 0 13 30 10 15 12 1 0 2 0 21 14 2 4 1 3 31 27 21 23 23 7 21 23 31 28 18 24 29 4 21 13 23 1 16 21 20 29 8 24 0 0 8 14 4 18 16 31 31 13 23 9 11 1 14 4 0 17 26 29 15 25 7 15 8 0 31 11 6 22 11 5 31 18 7 28 29 0 3 2 12 0 0 6 18 29 29 0 31 4 25 26 15 29 23 4 21 6 15 13 6 18 15 16 14 20 19 17 12 30 3 8 27 22 24 19 13 15 14 8 30 30 0 0 21 9 10 16 10 29 9 5 18 0 10 21 31 2 17 31 28 7 23 20 25 20 21 16 17 28 5 17 17 23 21 16 13 20 10 31 19 18 15 0 0 2 3 19 12 19 26 0 21 16 30 4 18 24 5 18 31 9 5 20 12 4 13 18 17 18 18 24 1 5 12 17 2 0 3 22 28 26 3 12 0 0 15 30 24 9 10 22 22 29 25 10 23 1 31 4 11 2 5 22 11 1 17 30 15 4 26 8 1 31 26 20 17 15 6 26 9 7 27 3 4 0 0 10 25 26 27 24 20 27 1 12 8 11 16 5 13 17 2 4 11 3 5 21 16 11 9 18 0 11 27 19 12 30 23 11 6 2 25 0 19 19 0 0 21 25 2 26 9 13 29 6 15 24 30 25 23 1 4 6 20 27 5 25 21 9 5 24 23 5 8 8 23 2 5 20 17 16 29 10 29 1 10 0 0 5 29 10 26 18 31 11 30 13 26 19 30 12 31 30 16 0 12 0 8 23 4 6 18 4 28 27 30 0 22 8 28 23 19 0 8 0 0 23 5 27 7 9 3 1 19 13 1 10 18 30 31 3 14 20 0 20 1 13 30 0 16 13 27 12 26 4 27 3 16 15 17 25 0 0 6 1 24 15 11 3 3 22 22 26 22 24 1 9 22 19 27 22 30 7 25 17 21 19 12 18 28 27 25 2 12 12 4 30 0 0 31 3 14 23 13 11 2 29 27 5 10 29 15 13 7 8 6 26 1 22 10 6 27 11 30 15 11 25 28 27 9 6 2 6 6 16 7 15 30 0 0 14 20 19 17 24 17 8 14 10 29 0 17 5 3 13 15 22 9 9 15 27 10 21 6 3 7 0 24 22 25 10 19 21 11 10 29 3 5 13 0 0 24 4 11 0 17 4 8 16 16 28 5 15 13 16 31 26 12 29 24 25 15 17 12 6 9 27 6 18 22 5 20 4 12 30 7 3 8 29 18 0 0 22 4 3 26 10 12 1 2 20 14 26 27 24 17 10 6 10 7 7 15 24 5 6 5 21 18 1 13 17 30 24 23 9 22 22 17 2 21 21 0 0 26 13 29 4 26 7 6 8 17 29 22 1 13 19 18 15 31 23 19 20 16 18 29 27 16 22 12 26 16 31 16 20 17 11 26 2 28 12 23 0 0 19 19 8 29 17 7 29 4 7 12 3 23 0 10 13 11 16 22 17 15 25 0 8 4 18 14 24 14 29 9 16 12 20 22 19 5 17 7 24 0 0 4 22 19 17 5 11 4 12 16 31 15 23 20 26 10 7 25 21 25 26 1 28 9 26 12 18 0 5 22 22 7 12 25 4 28 25 6 2 4 0 0 21 21 14 5 22 25 5 20 2 24 6 25 25 26 7 4 20 0 21 27 8 23 25 22 1 12 26 15 3 2 6 20 17 3 7 31 0 0 14 18 3 8 2 9 2 24 31 2 6 2 4 15 6 14 2 16 19 21 30 4 31 20 22 0 0 26 8 30 18 4 17 23 27 12 14 19 18 0 0 29 18 0 2 11 20 2 12 0 28 10 11 25 12 10 28 19 28 8 29 0 1 17 21 30 20 16 21 10 24 31 29 18 28 29 2 7 10 31 0 0 15 27 30 13 9 11 19 6 24 20 27 3 2 13 6 4 26 20 5 6 26 9 10 20 12 26 3 0 15 30 14 18 23 6 16 0 22 17 27 0 0 31 27 31 28 25 24 8 29 31 8 14 9 0 16 16 18 27 6 1 16 18 15 24 10 0 13 9 26 12 25 20 12 30 13 0 0 ==> Exponent value of non-zero matrix of 39-th row 3 11 13 7 0 ==> Exponent value of non-zero matrix of 40-th row 7 13 23 21 0 24 6 6 24 0 28 20 17 29 0 27 16 25 0 0 17 31 5 15 0 16 0 2 22 0 0 1 14 0 0 9 0 21 28 0 0 27 7 31 0 15 1 24 0 0 16 23 9 13 0 3 14 28 0 0 20 18 18 0 0 22 17 26 4 0 8 4 7 10 0 14 7 30 0 0 5 23 30 0 0 1 3 7 15 0 15 23 25 0 0 20 10 6 0 0 9 15 29 2 0 17 29 30 0 0 28 18 14 0 0 16 22 21 21 0 12 23 18 27 0 1 2 18 0 0 24 25 9 0 0 16 4 30 30 0 5 11 29 11 0 4 0 9 3 0 2 19 4 0 0 24 26 5 26 0 13 17 27 0 0 14 15 21 18 0 24 0 11 0 0 19 15 20 0 0 10 7 4 0 0 2 12 15 0 0 15 1 8 0 0 15 23 23 29 0 12 12 24 30 0 30 19 2 0 0 1 19 26 0 0 31 21 13 0 0 6 13 22 11 0 12 29 31 0 0 10 4 25 0 0 10 27 9 0 0 16 21 8 23 0 4 30 1 18 0 0 7 15 16 0 13 6 0 24 0 12 8 27 0 0 31 10 21 25 0 16 21 23 18 0 17 14 0 0 0 22 21 4 0 0 31 5 29 12 0 15 3 0 0 0 17 20 26 0 0 9 5 8 12 0 9 7 31 15 0 9 8 23 0 0 22 3 13 0 0 17 19 18 0 0 20 31 25 7 0 26 31 7 0 0 29 28 7 2 0 15 29 1 0 0 1 9 7 0 0 25 15 29 21 0 20 12 23 17 0 4 21 18 26 0 26 6 12 0 0 8 27 29 23 0 19 21 1 0 0 18 16 19 0 0 3 24 29 16 0 3 23 0 0 0 9 22 4 10 0 22 6 13 2 0 26 27 14 0 0 24 25 8 22 0 9 10 15 22 0 11 29 3 16 0 28 14 2 26 0 8 9 27 22 0 31 9 6 14 0 11 6 19 3 0 0 15 0 0 0 6 20 19 14 0 6 17 6 0 0 22 12 9 0 0 2 24 4 0 0 10 12 21 29 0 18 25 30 0 0 17 4 17 11 0 9 27 19 0 0 17 27 31 0 0 25 14 22 0 0 22 28 6 0 0 29 27 20 29 0 22 25 25 0 0 19 4 5 24 0 12 21 29 6 0 27 0 21 28 0 2 15 21 15 0 20 30 1 1 0 23 5 29 25 0 12 18 16 0 0 17 17 27 8 0 14 6 19 0 0 31 21 16 26 0 29 28 27 26 0 30 31 3 15 0 20 21 20 6 0 16 2 16 10 0 29 13 17 0 0 27 14 20 16 0 0 26 31 5 0 30 4 19 16 0 9 0 9 24 0 26 18 30 18 0 2 25 11 4 0 31 13 23 10 0 12 5 12 24 0 30 28 7 1 0 12 20 20 24 0 14 27 26 0 0 22 9 14 0 0 19 27 9 27 0 31 24 17 11 0 25 12 16 0 0 10 13 20 27 0 28 9 3 2 0 18 14 7 0 0 22 7 10 13 0 18 8 3 15 0 4 12 15 1 0 12 28 4 19 0 7 19 1 0 0 18 1 0 10 0 9 7 12 0 0 19 8 27 29 0 27 5 11 26 0 27 0 3 15 0 30 12 2 19 0 19 10 11 0 0 8 1 23 30 0 3 28 10 6 0 13 21 23 19 0 9 22 23 11 0 23 13 11 28 0 24 8 30 22 0 19 3 11 23 0 23 30 0 0 0 6 2 19 1 0 4 2 6 16 0 0 26 30 0 0 11 10 7 0 0 19 22 10 3 0 8 27 18 10 0 26 15 17 28 0 25 23 10 2 0 24 10 4 0 0 19 0 25 3 0 25 10 12 23 0 16 12 15 19 0 15 18 14 0 0 11 17 28 0 0 19 24 25 26 0 29 17 9 0 0 21 2 26 8 0 17 13 3 17 0 5 1 16 13 0 7 18 23 8 0 10 21 10 29 0 5 22 21 0 0 19 22 9 10 0 12 7 3 27 0 27 15 9 8 0 22 28 9 0 0 26 13 15 8 0 15 26 9 0 0 27 23 25 5 0 4 29 22 5 0 18 22 20 18 0 14 20 28 12 0 22 9 2 8 0 12 24 17 0 0 8 26 9 0 0 29 31 28 3 0 29 8 26 1 0 25 23 16 10 0 7 28 13 20 0 22 30 5 17 0 12 16 8 16 0 0 1 14 14 0 26 31 10 2 0 6 8 26 3 0 3 11 14 0 0 25 17 23 7 0 20 13 9 13 0 18 20 31 0 0 28 13 5 0 0 3 3 18 0 0 18 20 12 24 0 29 27 21 17 0 2 21 16 18 0 16 31 23 28 0 23 13 28 17 0 5 26 20 0 0 20 18 22 5 0 8 5 9 0 0 18 12 12 0 0 7 8 26 0 0 10 3 19 16 0 7 8 17 26 0 29 0 18 25 0 4 0 28 2 0 23 5 25 18 0 7 30 6 8 0 29 24 6 29 0 2 25 3 0 0 12 24 16 25 0 17 19 6 11 0 31 10 28 17 0 20 21 11 1 0 10 30 23 31 0 12 16 5 8 0 0 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0 28 8 28 10 0 14 9 15 11 0 1 15 6 10 0 21 5 14 0 0 22 7 7 4 0 3 18 17 0 0 2 21 22 22 0 30 27 24 21 0 15 30 10 9 0 18 26 13 0 0 0 12 12 0 0 21 15 16 0 0 12 12 4 0 0 4 29 25 0 0 28 31 3 2 0 22 21 20 0 0 15 7 22 18 0 17 15 20 24 0 3 12 4 19 0 16 14 7 19 0 21 10 9 0 0 2 8 19 0 0 9 20 26 0 0 6 27 0 0 0 25 5 10 14 0 25 7 2 17 0 19 13 13 23 0 14 24 27 30 0 30 25 8 9 0 10 9 25 20 0 18 26 7 0 0 4 17 13 0 0 31 2 31 0 0 6 9 10 26 0 30 8 31 7 0 23 1 2 0 0 19 21 31 2 0 31 10 22 0 0 22 27 18 0 0 7 9 17 0 0 1 4 20 6 0 26 21 19 29 0 1 25 17 29 0 19 17 5 19 0 21 25 1 27 0 10 24 8 27 0 9 27 1 0 0 16 14 29 9 0 14 16 9 0 0 13 2 19 0 0 3 24 26 31 0 4 10 7 16 0 4 10 2 15 0 29 30 13 1 0 14 24 27 31 0 29 18 18 6 0 1 30 16 15 0 30 23 12 2 0 18 28 30 6 0 16 30 2 0 0 28 11 24 0 0 25 13 21 30 0 17 1 8 25 0 6 18 5 18 0 8 0 28 0 0 13 31 7 11 0 25 12 13 0 0 10 5 18 0 0 0 6 21 0 0 21 31 12 20 0 2 24 9 0 0 12 15 18 16 0 20 4 12 27 0 9 26 6 21 0 1 3 30 24 0 28 11 30 18 0 27 29 19 14 0 22 9 2 16 0 8 4 22 1 0 14 16 29 0 0 12 16 30 19 0 7 31 17 31 0 22 0 5 29 0 12 7 16 0 0 28 20 7 7 0 9 16 9 4 0 11 31 28 7 0 16 4 12 27 0 8 5 22 0 0 27 17 17 26 0 20 18 5 18 0 12 13 17 3 0 3 14 22 17 0 25 9 19 0 0 23 17 3 0 0 1 23 20 0 0 15 27 15 0 0 25 28 2 0 0 13 3 13 0 0 0 5 11 0 0 9 11 0 0 0 20 13 23 13 0 30 1 0 26 0 26 0 8 21 0 18 29 2 0 0 18 25 14 0 0 31 16 1 0 0 13 30 31 15 0 29 6 31 11 0 16 18 10 28 0 8 12 20 0 0 19 29 29 18 0 6 31 3 26 0 1 29 4 9 0 24 5 3 0 0 10 0 12 17 0 21 30 16 0 0 5 23 0 28 0 15 17 11 12 0 9 3 21 0 0 18 15 10 20 0 24 17 7 22 0 7 6 2 19 0 7 4 15 31 0 8 13 16 4 0 1 11 27 16 0 23 0 29 11 0 0 23 31 12 0 8 31 8 12 0 19 25 4 27 0 4 16 4 17 0 30 9 5 8 0 9 27 21 8 0 24 0 19 0 0 3 8 12 23 0 3 11 30 18 0 0 6 21 23 0 22 23 4 0 0 26 22 28 9 0 23 28 16 0 0 27 22 1 23 0 2 14 18 0 0 28 11 8 1 0 24 28 6 0 0 26 22 5 3 0 25 21 0 0 0 ==> Exponent value of non-zero matrix of 1534-th row 24 28 26 0 0 ==> Exponent value of non-zero matrix of 1535-th row

FIGS. 4 to 8 show examples of the PCM structure of the LDPC code according to embodiments of the present invention.

As an example, FIG. 4 illustrates the PCM structure of a QC-LDPC code for terrestrial cloud broadcast, wherein the length of the codeword is 8192, and FIG. 5 is an expanded view of the PCM structure shown in FIG. 4.

Meanwhile, FIG. 6 illustrates the PCM structure of a QC-LDPC code for terrestrial cloud broadcast, wherein the length of the codeword is 16384, FIG. 7 illustrates the PCM structure of a QC-LDPC code for terrestrial cloud broadcast, wherein the length of the codeword is 32768, and FIG. 8 illustrates the PCM structure of a QC-LDPC code for terrestrial cloud broadcast, wherein the length of the codeword is 65536.

FIG. 9 illustrates an example of the PCM structure of an LDPC code according to code rates according to an embodiment of the present invention.

The PCM of the LDPC code according to the present invention consists of multiple single parity check codes, and thus, it may be truncated for each of the different code rates, as shown in FIG. 9. For example, LDPC codes having code rates of ½ and ⅓ may be easily generated by truncating the PCM by 50 and 83.3%. This means that only part of the overall codeword may be decoded in a high SNR region by the LDPC code according to the present invention. Accordingly, the LDPC decoder may save power consumption and reduce waiting time.

FIG. 10 is a graph illustrating the performance of LDPC codes according to an embodiment of the present invention.

As an example, in FIG. 10, the performance of the QC-LDPC code having a code rate of 0.25 is shown as compared with the SNR. For purposes of computational experiment, an LLR (Log-likelihood Ratio)-based sum-product algorithm has been assumed that performs QPSK (Quadratic Phase Shift Keying) modulation and 50 counts of repeated decoding.

Meanwhile, FIG. 10 also illustrates the LDPC codes used in the DVB-T2/S2 system, which have a code rate of 0.25 and a codeword length of 64800, to show the excellent performance of the LDPC codes newly designed according to the present invention. Further, Table 2 below shows the performance gap that stems from the Shannon limit of the LDPC code designed for terrestrial cloud broadcast at BER (Bit Error Probability)=2×10⁻⁶ and Table 3 shows the complexity of the LDPC code that is proportional to the number of 1's in the PCM.

TABLE 2 Shannon Limit Gap from Limit Length (N) [SNR, dB] [dB]  8192 −3.804 1.29 16384 −3.804 0.99 32768 −3.804 0.79 65536 −3.804 0.6 DVB (64800) −3.804 1.29

TABLE 3 Number of ones in Length (N) PCM  8192 36,352 16384 72,736 32768 145,504 65536 291,040 DVB (64800) 194,399

Referring to FIG. 10 and Tables 2 and 3, among the LDPC codes newly designed according to the present invention, the LDPC code having a codeword length of 65536 is about 0.69 dB better in performance but about 50% higher in complexity than the LDPC code of the DVB-T2/S2 system, which has a codeword length of 64800. However, the LDPC codes having the codeword lengths of 16384 and 32768 are respectively more excellent in performance by about 0.3 dB and about 0.5 dB and lower in complexity by about 63% and about 25% than the LDPC code of the DVB-T2/S2 system which has a codeword length of 64800. Further, the LDPC codes newly designed according to the present invention, when their codeword lengths are 8192, 16384, 32768, and 65536, are spaced apart from the Shannon limit by about 1.29 dB, about 0.99 dB, about 0.79 dB, and about 0.6 dB at BER=2×10⁻⁶.

FIG. 11 is a flowchart illustrating a method of decoding an LDPC codeword according to an embodiment of the present invention.

Referring to FIG. 11, the LDPC decoder according to the present invention, when receiving an LDPC codeword (1110), decodes the received LDPC codeword using a parity check matrix obtained by combining a first parity check matrix for an LDPC code having a higher code rate than a reference value (e.g., 0.5) with a second parity check matrix for an LDPC code having a lower code word than the reference value (1120). At this time, the parity check matrix, as shown in FIG. 3, may include the zero matrix Z, the identity matrix D, and the dual diagonal matrix B. In the dual diagonal matrix B, the element matrix constituting the dual diagonal line is an identity matrix, and the remaining element matrixes may be zero matrixes. The element matrix constituting the dual diagonal line of the diagonal matrix B may be continuous to the element matrix constituting the diagonal line of the identity matrix D.

FIG. 12 is a block diagram illustrating an LDPC encoder and an LDPC decoder according to an embodiment of the present invention.

Referring to FIG. 12, the LDPC encoder 1210 according to the present invention may include, as an example, an input unit 1212, a determining unit 1214, and an encoding unit 1216. The LDPC decoder 1220 may include a receiving unit 1222 and a decoding unit 1224.

The input unit 1212 receives information to be encoded.

The determining unit 1214 determines a code rate of the LDPC code and determines the size of a dual, diagonal matrix depending on the determined code rate.

The encoding unit 1216 encodes the information input through the input unit 1212 with the LDPC codeword by using the parity check matrix having the code rate determined by the determining unit 1214. Here, the parity check matrix may consist of zero matrixes and circulant permutation matrixes.

Meanwhile, the parity check matrix, as an example, may have the structure in which a first parity check matrix for an LDPC code having a higher code rate than a reference value and a second parity check matrix for an LDPC code having a lower code rate than the reference value. Here, the reference value may be, e.g., 0.5. It may include a zero matrix, an identity matrix, and a dual diagonal matrix. Specifically, the parity check matrix may have a structure as shown in FIG. 3, and in such case, the first parity check matrix may include the dual diagonal matrix B, and the second parity check matrix may include the identity matrix D. In the dual diagonal matrix B, the element matrix constituting the dual diagonal line may be an identity matrix while the remaining element matrixes may be zero matrixes. The element matrix constituting the dual diagonal line may be continuous to the element matrix constituting the diagonal line of the identity matrix D.

The encoding unit 1216, as an example, may encode the input information with the LDPC codeword by calculating the first parity part using the information input through the input unit 1212 and the first parity check matrix and by calculating the second parity part using the second parity check matrix based on the input information and the calculated first parity part.

The LDPC codeword encoded by the encoding unit 1216 may include the systematic part corresponding to the information input through the input unit 1212, the first parity part corresponding to the dual diagonal matrix, and the second parity part corresponding to the identity matrix.

Meanwhile, the receiving unit 1222 of the decoder 1220 receives the LDPC codeword encoded by the parity check matrix. The LDPC codeword received by the receiving unit 1222 may include the systematic part, the first parity part, and the second parity part.

The decoding unit 1224, as shown in FIG. 3, decodes the LDPC codeword received by the receiving unit 1222 using the parity check matrix as shown in FIG. 3.

Although exemplary embodiments of the present invention have been described, the present invention is not limited thereto, and various modifications or variations may be made thereto without departing from the scope of the present invention. The embodiments described herein are not provided to limit the present invention but to describe the invention, and the present invention is not limited thereto. The scope of the present invention should be interpreted within the appended claims and the spirit within the equivalents of the invention should be construed to be included in the scope of the invention. 

What is claimed is:
 1. A method of decoding an LDPC code by an LDPC (Low Density Parity Check) decoder, the method comprising: receiving an LDPC codeword encoded by a parity check matrix; and decoding the received LDPC codeword using the parity check matrix, wherein the parity check matrix includes a first matrix, a dual diagonal matrix which has the same number of rows and columns as the number of rows of the first matrix, a zero matrix which has the same number of rows as the number of columns of the dual diagonal matrix, a second matrix which has the same number of rows as the number of the columns of the zero matrix, and an identity matrix which has the same number of rows and columns as the number of the columns of the zero matrix, wherein a last column of the first matrix is adjacent to a first column of the dual diagonal matrix in the parity check matrix, a last row of the first matrix and a last row of the dual diagonal matrix are adjacent to a first row of the second matrix in the parity check matrix, and wherein an element matrix of the dual diagonal matrix constituting a dual diagonal line is continuous to an element matrix that constitutes a diagonal line of the identity matrix.
 2. The method of claim 1, the identity matrix is adjacent to the second matrix and the zero matrix.
 3. The method of claim 2, wherein the encoded LDPC codeword includes a systematic part corresponding to the input information, a first parity part corresponding to the dual diagonal matrix, and a second parity part corresponding to the identity matrix.
 4. The method of claim 2, wherein the parity check matrix includes a circulant permutation matrix.
 5. An LDPC (Low Density Parity Check) decoder comprising: a receiving unit for receiving an LDPC codeword encoded by a parity check matrix; and a decoding unit for decoding the received LDPC codeword using the parity check matrix, wherein the parity check matrix includes a first matrix, a dual diagonal matrix which has the same number of rows and columns as the number of rows of the first matrix, a zero matrix which has the same number of rows as the number of columns of the dual diagonal matrix, a second matrix which has the same number of rows as the number of the columns of the zero matrix, and an identity matrix which has the same number of rows and columns as the number of the columns of the zero matrix, wherein a last column of the first matrix is adjacent to a first column of the dual diagonal matrix in the parity check matrix, a last row of the first matrix and a last row of the dual diagonal matrix are adjacent to a first row of the second matrix in the parity check matrix, and wherein an element matrix of the dual diagonal matrix constituting a dual diagonal line is continuous to an element matrix that constitutes a diagonal line of the identity matrix.
 6. The LDPC decoder of claim 5, the identity matrix is adjacent to the second matrix and the zero matrix.
 7. The LDPC decoder of claim 6, wherein the encoded LDPC codeword includes a systematic part corresponding to the input information, a first parity part corresponding to the dual diagonal matrix, and a second parity part corresponding to the identity matrix.
 8. The LDPC decoder of claim 6, wherein the parity check matrix includes a circulant permutation matrix. 